SymFun 3.1: Documentationww3.haverford.edu/math/cgreene/symfun/symfun31-doc.pdfSymFun 3.1:...

SymFun 3.1: Documentation

Summer 2004: Created by Curtis Greene, Julien Colvin, Ben FinemanSummer 2009: Updates by Curtis Greene, Rengyi XuSummer 2010: Updates by Ian BurnetteSummer 2012: Updates by Curtis Greene

IntroductionThis is a Mathematica package designed to support various manipulations of symmetric functions, with anemphasis on combinatorial applications. It is loosely modeled after John Stembridge's highly successful Maplepackage ("SF"), but it is somewhat less ambitious: currently it implements only the standard basis conversions anda few other operations such as the Littlewood-Richardson rule.

We assume familiarity with the basic theory of symmetric functions, e.g. Macdonald' s Symmetric Functions andHall Polynomials or Stanley' s EC2.

Using the PackageFind the package (symfun31.m), open it, and click the button labeled "Run Package". Alternatively, load thepackage using the command

<< symfun31.m

In order for the latter method to work, the package must be in a location recognized by Mathematica's $Pathvariable, for example,

"C:\\Program Files\\Wolfram Research\\Mathematica\\7.0\\AddOns\\ExtraPackages"

Overview: Conventions, Notation, and Basic OperationsBasic families of symmetric functions (elementary, homogeneous, monomial, power sum, and Schur) may bedefined in several ways. The following are expressions that are "atomic", i.e., they are unevaluated by Mathemat-ica and hence treated as indeterminates:

e@4D,h@5D,p@3D,m@3,2,1D,s@3,2,1D

The first three combine to give monomials in the obvious way, e.g.,

e@3De@1D^2, h@4Dh@2Dh@1D,p@3Dp@2Dp@1D

These are the standard forms for the elementary, homogenous, and power-sum symmetric function. Expressionsof the form

e@83,1,1<D, h@84,2,1<D, p@83,2,1<D

are evaluated by Mathematica, and give equivalent e, h, and p-functions in the above monomial form.

Appending “x” to the function name and including an “n” parameter yields an evaluation in n variables x[1],x[2],...,x[n], for example,

ex@83, 1, 1<, 3D

x@1D3 x@2D x@3D + 2 x@1D2 x@2D2 x@3D + x@1D x@2D3 x@3D +

2 x@1D2 x@2D x@3D2+ 2 x@1D x@2D2 x@3D2

+ x@1D x@2D x@3D3

px@83, 1, 1<, 3D

x@1D5+ 2 x@1D4 x@2D + x@1D3 x@2D2

+ x@1D2 x@2D3+ 2 x@1D x@2D4

+ x@2D5+ 2 x@1D4 x@3D +

2 x@1D3 x@2D x@3D + 2 x@1D x@2D3 x@3D + 2 x@2D4 x@3D + x@1D3 x@3D2+ x@2D3 x@3D2

x@1D2 x@3D3+ 2 x@1D x@2D x@3D3

+ x@2D2 x@3D3+ 2 x@1D x@3D4

+ 2 x@2D x@3D4+ x@3D5

The same result can be obtained using "conversion" commands (described fully later in this document), e.g.

EToX@e@83, 1, 1<D, 3D

2 x@1D2 x@2D x@3D2+ 2 x@1D x@2D2 x@3D2

+ x@1D x@2D x@3D3

EToX@e@3D e@1D^2, 3D

2 x@1D2 x@2D x@3D2+ 2 x@1D x@2D2 x@3D2

+ x@1D x@2D x@3D3

Conversions are available between all pairs of families, e.g.,

EToS@e@3D e@1D^2D

s@2, 2, 1D + s@3, 1, 1D + 2 s@2, 1, 1, 1D + s@1, 1, 1, 1, 1D

EToP@e@3D e@1D^2D

2p@1D3 p@2D +

3p@1D2 p@3D

Certain other algebraic operations are computed automatically. For example, products of atomic expressionsinvolving the Schur functions are automatically evaluated using the Littlewood-Richardson rule.

s@3, 2, 1D s@2, 1D

s@3, 3, 3D + 2 s@4, 3, 2D + s@4, 4, 1D + s@5, 2, 2D + s@5, 3, 1D +

s@3, 2, 2, 2D + 2 s@3, 3, 2, 1D + 2 s@4, 2, 2, 1D + 2 s@4, 3, 1, 1D +

s@5, 2, 1, 1D + s@3, 2, 2, 1, 1D + s@3, 3, 1, 1, 1D + s@4, 2, 1, 1, 1D

Basic Commands: Entering and Manipulating Symmetric Functions� H: definitions

h@<Λ>D

Expands in terms of h[1],h[2],h[3],...

h@83, 3, 2<D

h@2D h@3D2

hx@<lambda>,<n>D

Returns h[Λ] expanded in terms of x[1],x[2],...x[n] .

2 symfun31-doc.nb

hx@82<, 3D

x@1D2+ x@1D x@2D + x@2D2

+ x@1D x@3D + x@2D x@3D + x@3D2

hx@83, 2<, 2D

x@1D5+ 2 x@1D4 x@2D + 3 x@1D3 x@2D2

+ 3 x@1D2 x@2D3+ 2 x@1D x@2D4

+ x@2D5

he@<n>D

Expands h[n] directly into the e’s. This syntax works for the "primitive" h, e, and p functions, i.e. they can beconverted directly to the others.

e@1D5- 4 e@1D3 e@2D + 3 e@1D e@2D2

+ 3 e@1D2 e@3D - 2 e@2D e@3D - 2 e@1D e@4D + e@5D

12p@1D3 p@2D +

8p@1D p@2D2

6p@1D2 p@3D +

6p@2D p@3D +

4p@1D p@4D +

HList@<n>D

Returns list of all h[Λ]'s of degree n, expressed as monomials in h[1],h[2],... (Useful in some technical situations.)

HList@5D

9h@5D, h@1D h@4D, h@2D h@3D, h@1D2 h@3D, h@1D h@2D2, h@1D3 h@2D, h@1D5=

HList@<n>, <k>D

Returns list of all h[Λ]'s of degree n and Length(Λ)£k, expressed as monomials in h[1],h[2],...

HList@5, 2D

8h@5D, h@1D h@4D, h@2D h@3D<

HList2@<n>, <k>D

Returns list of all h[Λ]'s of degree n and Λ[[1]]£k, expressed as monomials in h[1],h[2],...

HList2@5, 2D

9h@1D h@2D2, h@1D3 h@2D, h@1D5=

� E: definitions

e@<Λ>D

Expands in terms of e[1],e[2],e[3],...

e@83, 2, 2, 1<D

e@1D e@2D2 e@3D

ex@<Λ>, <n>D

Returns e[Λ] expanded in terms of x[1],x[2],...,x[n] . Similar construction works for e, p, and h.

ex@83, 2<, 3D

x@1D2 x@2D2 x@3D + x@1D2 x@2D x@3D2+ x@1D x@2D2 x@3D2

eh@<n>D

Returns e[n] expressed in terms of h[1],h[2],.... . Similar construction works for e, p, and h.

symfun31-doc.nb 3

h@1D5- 4 h@1D3 h@2D + 3 h@1D h@2D2

+ 3 h@1D2 h@3D - 2 h@2D h@3D - 2 h@1D h@4D + h@5D

12p@1D3 p@2D +

8p@1D p@2D2

6p@1D2 p@3D -

6p@2D p@3D -

4p@1D p@4D +

EList@<n>D

Returns list of all e[lambda]'s of degree n, expressed as monomials in e[1],e[2],... (Useful in some technicalsituations.)

EList@6D

9e@6D, e@1D e@5D, e@2D e@4D, e@1D2 e@4D, e@3D2,

e@1D e@2D e@3D, e@1D3 e@3D, e@2D3, e@1D2 e@2D2, e@1D4 e@2D, e@1D6=

EList@<n>, <k>D

Returns list of all e[lambda]'s of degree n and lambda[[1]]£k, expressed as monomials in e[1],e[2],...

EList@6, 2D

9e@1D6, e@1D4 e@2D, e@1D2 e@2D2, e@2D3=

� M: definitions

m@<Λ>D

Converts m[{Λ1,Λ2,...}] notation into m[Λ1,Λ2,...]. Useful for mapping m onto a list of partitions.

m@83, 2, 1<D

m@3, 2, 1D

monomial@<exponentlist>D

Returns a single monomial of type <exponentlist>.

monomial@82, 4, 3, 1<D

x@1D2 x@2D4 x@3D3 x@4D

mx@<Λ>, <n>D

Returns m[Λ] expanded in terms of x[1],x[2],...,x[n].

mx@83, 2, 1<, 3D

x@1D3 x@2D2 x@3D + x@1D2 x@2D3 x@3D + x@1D3 x@2D x@3D2+

x@1D x@2D3 x@3D2+ x@1D2 x@2D x@3D3

+ x@1D x@2D2 x@3D3

MList@<n>D

Returns list of all m[Λ]'s of degree n. (Useful in some technical situations.)

MList@4D

8m@4D, m@3, 1D, m@2, 2D, m@2, 1, 1D, m@1, 1, 1, 1D<

MList@<n>, <k>D

Returns list of all m[Λ]'s of degree n with Length(Λ)£k

4 symfun31-doc.nb

MList@4, 3D

8m@4D, m@3, 1D, m@2, 2D, m@2, 1, 1D<

� P: definitions

p@<Λ>D

Expands in terms of p[1],p[2],p[3],...

p@83, 2, 2, 1<D

p@1D p@2D2 p@3D

px@<Λ>,<n>D

Returns p[lambda] expanded in terms of x[1],x[2],...,x[n].

px@83, 2, 2, 1<, 2D

x@1D8+ x@1D7 x@2D + 2 x@1D6 x@2D2

+ 3 x@1D5 x@2D3+

2 x@1D4 x@2D4+ 3 x@1D3 x@2D5

+ 2 x@1D2 x@2D6+ x@1D x@2D7

+ x@2D8

px@84<, 6D

x@1D4+ x@2D4

+ x@3D4+ x@4D4

+ x@5D4+ x@6D4

pe@<n>D

Returns p[n] expanded in terms of e[1],e[2],... Similar construction works for p, e, and h.

e@1D4- 4 e@1D2 e@2D + 2 e@2D2

+ 4 e@1D e@3D - 4 e@4D

-h@1D4+ 4 h@1D2 h@2D - 2 h@2D2

- 4 h@1D h@3D + 4 h@4D

PList@<n>D

Returns list of all p[Λ]'s of degree n,expressed as monomials in p[1],p[2],... (Useful in some technical situations.)

PList@5D

9p@5D, p@1D p@4D, p@2D p@3D, p@1D2 p@3D, p@1D p@2D2, p@1D3 p@2D, p@1D5=

PList@<n>, <k>D

Returns list of all p[Λ]'s of degree n, and Length[Λ]£k, expressed as monomials in p[1],p[2],...

PList@5, 2D

8p@5D, p@1D p@4D, p@2D p@3D<

� S: definitions

s@<Λ>D

Converts s[{Λ1,Λ2,...}] notation into s[Λ1,Λ2,...]. Useful for mapping m onto a list of partitions.

s@84, 2, 1, 1<D

s@4, 2, 1, 1D

sx@<Λ>,<n>D

Returns s[Λ] expanded in terms of x[1],x[2],...x[n] .

symfun31-doc.nb 5

sx@84, 2, 1<, 3D

x@1D4 x@2D2 x@3D + x@1D3 x@2D3 x@3D + x@1D2 x@2D4 x@3D + x@1D4 x@2D x@3D2+

2 x@1D3 x@2D2 x@3D2+ 2 x@1D2 x@2D3 x@3D2

+ x@1D x@2D4 x@3D2+ x@1D3 x@2D x@3D3

2 x@1D2 x@2D2 x@3D3+ x@1D x@2D3 x@3D3

+ x@1D2 x@2D x@3D4+ x@1D x@2D2 x@3D4

sh@<Λ>D

Returns s[Λ] in terms of a linear combination of h[1], h[2], ... . Also works if argument is a sequence, or a singleinteger.

sh@84, 2, 1<Dsh@4, 2, 1Dsh@4D

h@1D h@2D h@4D - h@3D h@4D - h@1D2 h@5D + h@1D h@6D

h@1D h@2D h@4D - h@3D h@4D - h@1D2 h@5D + h@1D h@6Dh@4D

se@<Λ>D

Returns s[Λ] in terms of a linear combination of e[1], e[2], ... Also works if argument is a sequence, or a singleinteger.

se@84, 2, 1<Dse@4, 2, 1Dse@4D

e@1D2 e@2D e@3D - e@2D2 e@3D - e@1D e@3D2-

e@1D3 e@4D + e@1D e@2D e@4D + e@3D e@4D + e@1D2 e@5D - e@1D e@6D

e@1D2 e@2D e@3D - e@2D2 e@3D - e@1D e@3D2-

e@1D3 e@4D + e@1D e@2D e@4D + e@3D e@4D + e@1D2 e@5D - e@1D e@6D

e@1D4- 3 e@1D2 e@2D + e@2D2

+ 2 e@1D e@3D - e@4D

SList@<n>D

Returns list of all s[Λ]'s of degree n. (Useful in some technical situations.)

SList@7D

8s@7D, s@6, 1D, s@5, 2D, s@5, 1, 1D, s@4, 3D, s@4, 2, 1D,

s@4, 1, 1, 1D, s@3, 3, 1D, s@3, 2, 2D, s@3, 2, 1, 1D, s@3, 1, 1, 1, 1D,

s@2, 2, 2, 1D, s@2, 2, 1, 1, 1D, s@2, 1, 1, 1, 1, 1D, s@1, 1, 1, 1, 1, 1, 1D<SList@<n>, <k>D

Returns list of all s[Λ]'s of degree n with Length(Λ)£k

SList@7, 3D

8s@7D, s@6, 1D, s@5, 2D, s@5, 1, 1D, s@4, 3D, s@4, 2, 1D, s@3, 3, 1D, s@3, 2, 2D<

Basic Commands: Partitions, Degrees, Coefficients, Etc.� Partitions

ConjugateP@<Λ>D

6 symfun31-doc.nb

Computes the conjugate of a partition Λ.

ConjugateP@84, 2, 1, 1, 1, 1, 1, 1<D

88, 2, 1, 1<

Partitions@<n>,@ Rows® r,Columns®cDD

Returns a list of partitions of n with £ r parts and largest part b c. Either, both, or none of the options is possible.(Extension of Combinatorica's Partition command.)

Partitions@7, Rows ® 3D

887<, 86, 1<, 85, 2<, 85, 1, 1<, 84, 3<, 84, 2, 1<, 83, 3, 1<, 83, 2, 2<<

Partitions@7, Columns ® 3D

883, 3, 1<, 83, 2, 2<, 83, 2, 1, 1<, 83, 1, 1, 1, 1<,

82, 2, 2, 1<, 82, 2, 1, 1, 1<, 82, 1, 1, 1, 1, 1<, 81, 1, 1, 1, 1, 1, 1<<Partitions@7, Rows ® 3, Columns ® 3D

883, 3, 1<, 83, 2, 2<<

� Degrees in a polynomial

MonDegrees@<f>,<variable list>D

Returns a sorted list of all of the total degrees of monomials (in the listed variables) appearing in f.

f = Product@1 - x@iD^2, 8i, 1, 3<D �� Expand

MonDegrees@f, 8x@1D, x@2D, x@3D<D

1 - x@1D2- x@2D2

+ x@1D2 x@2D2- x@3D2

+ x@1D2 x@3D2+ x@2D2 x@3D2

- x@1D2 x@2D2 x@3D2

86, 4, 2, 0<

MonDegrees@1 + x + y^2 + x y^3 + z^5, 8x, y, z<DMonDegrees@1 + x + y^2 + x y^3 + z^5, 8y, z<DMonDegrees@1 + x + y^2 + x y^3 + z^5, 8z<D85, 4, 2, 1, 0<

85, 3, 2, 0<

85, 0<

XDegrees@<f>D

Returns a sorted list of all of the degrees of in x (assumed) appearing in f. Useful when the number of variables isunknown.

XDegrees@x@1D x@2D^2 x@3D - 43 x@3424D + 321 x@40D^329 x@47D^32 x@412DD

8362, 4, 1<

XDegrees@1 + x@1D + x@1D x@2DD

82, 1, 0<

SDegrees@<f>D

Returns a sorted list of all of the degrees of s's appearing in f (assuming f is expressed in terms of s's).

SDegrees@6 * s@4, 3, 2, 1D + s@3, 1, 1DD

810, 5<

symfun31-doc.nb 7

SDegrees@9 + 6 s@4, 3, 2, 1D + s@3, 1, 1DDSDegrees@6 s@4, 3, 2, 1D + s@3, 1, 1D + x + 9D810, 5, 0<

810, 5, 0, x<

MDegrees@<f>D

Returns a sorted list of all of the degrees of m's appearing in f (assuming f is expressed in terms of m's).

MDegrees@5 * m@4, 2D - 4 m@1D - 7 m@2, 2, 2, 2DD

88, 6, 1<

EDegrees@<f>D

Returns a sorted list of all of the degrees of e's appearing in f (assuming f is expressed in terms of e's).

EDegrees@6 e@4D^4 e@2D e@1D^3 + e@4DD

821, 4<

HDegrees@<f>D

Returns a sorted list of all of the degrees of h's appearing in f (assuming f is expressed in terms of h's).

HDegrees@h@4D h@2D h@1D^2 - 4 h@3DD

88, 3<

PDegrees@<f>D

Returns a sorted list of all of the degrees of p's appearing in f (assuming f is expressed in terms of p's).

PDegrees@p@4D p@2D p@1D^2 - 4 p@3DD

88, 3<

� Coefficients

GetCoefficients@<f>, <variable list>D

Returns a list of the coefficients in f of the variables in <variable list>. The expression f should be a polynomial insubscripted variables, e.g., x[i]'s, h[Λ]'s, s[Λ],etc. If a monomial in f divides another monomial, only the formercoefficient is returned.

SList@5DGetCoefficients@s@3, 2D - 6 s@3, 1, 1D + 93 s@2, 1, 1, 1D - s@5D, SList@5DD8s@5D, s@4, 1D, s@3, 2D, s@3, 1, 1D, s@2, 2, 1D, s@2, 1, 1, 1D, s@1, 1, 1, 1, 1D<

8-1, 0, 1, -6, 0, 93, 0<

g = 7 e@2D^2 e@1D - e@4D e@2D e@1D - 6 e@3D^2 e@1D -

2 e@4D e@1D + 200 e@1D^5 - 3 e@7D - 81 e@2D e@1D^5 + 8 e@5D;

GetCoefficients@g, EList@5DDGetCoefficients@g, EList@7DDGetCoefficients@g, EList@3DD88, -2, 0, 0, 7, 0, 200<

8-3, 0, 0, 0, 0, -1, 0, -6, 0, 0, 0, 0, 0, -81, 0<

80, 0, 0<

GetCoefficients@g^3 �� Expand, EList@21DD

8 symfun31-doc.nb

� Symmetry tests, symmetrization

SymmetricQ@<f>,<z>,<n>D

Tests whether a polynomial in variables z[1],z[2],...,z[n] is symmetric.

SymmetricQ@Hx@1D + x@2D + x@3D + x@4D + x@5DL^10, x, 5D

SymmetricQ@Hx@1D + x@2D + x@3D + x@4D + x@5DL^10, x, 6D

Symmetrize@<f>,<x>,<n>D

Computes ÚΣ f Σ over all permutations Σ of x[1],x[2],...,x[n].

f = x@1D^2 + x@2D^2 ;

Symmetrize@f, x, 5DSymmetricQ@%, x, 5D

48 x@1D2+ 48 x@2D2

+ 48 x@3D2+ 48 x@4D2

+ 48 x@5D2

� Restrict number of variables

Restrict@<f>,<n>, <opts>D

Takes the symmetric function f and returns the equivalent function when the underlying x-version is restricted to nvariables.

This example takes a function in the h - basis, converts it to the s - basis, then restricts the number of variables to3, eliminating the s - terms with more than 3 parts.

f = h@1D^6 - 3 h@3D^2 + h@2D^3 - 5 h@5D h@1Dr1 = HToS@fDr1 = Restrict@r1, 3D

h@1D6+ h@2D3

- 3 h@3D2- 5 h@1D h@5D

-6 s@6D + 3 s@3, 3D + 9 s@4, 2D - s@5, 1D + 6 s@2, 2, 2D + 18 s@3, 2, 1D + 11 s@4, 1, 1D +

9 s@2, 2, 1, 1D + 10 s@3, 1, 1, 1D + 5 s@2, 1, 1, 1, 1D + s@1, 1, 1, 1, 1, 1D-6 s@6D + 3 s@3, 3D + 9 s@4, 2D - s@5, 1D + 6 s@2, 2, 2D + 18 s@3, 2, 1D + 11 s@4, 1, 1D

The same result could have been obtained directly using the HToS conversion command with the option Vars ->3.

r2 = HToS@f, Vars ® 3Dr1 � r2

-6 s@6D + 3 s@3, 3D + 9 s@4, 2D - s@5, 1D + 6 s@2, 2, 2D + 18 s@3, 2, 1D + 11 s@4, 1, 1D

Note that the results may be complex in the case of the h - and p - bases. In the other cases, the command operatesmore simply on individual monomials.

f = h@6DRestrict@f, 4Dh@6D

-h@1D6+ 3 h@1D4 h@2D - h@2D3

- 2 h@1D3 h@3D - 2 h@1D h@2D h@3D + h@3D2+ h@1D2 h@4D + 2 h@2D h@4D

symfun31-doc.nb 9

f = p@6DRestrict@f, 4Dp@6D

24p@1D6

8p@1D4 p@2D -

8p@1D2 p@2D2

3p@1D3 p@3D +

4p@1D2 p@4D +

4p@2D p@4D

Input to the Restrict command can be a function expressed using more than one basis. The Bases option allowsthe user to specify which bases the variable-restriction gets applied to. (Admittedly, this is not likely to be veryuseful!).

fmixed = e@4D e@2D + p@3D^2 + p@5D + x@1D x@2D x@3D + x@1D x@2D x@4D +

x@1D x@3D x@4D + x@2D x@3D x@4D + 4 m@2, 2D + s@3, 1, 1, 1, 1, 1, 1, 1, 1D - h@7DRestrict@fmixed, 3DRestrict@fmixed, 3, Bases ® 8E, P, X<D

e@2D e@4D - h@7D + 4 m@2, 2D + p@3D2+ p@5D + s@3, 1, 1, 1, 1, 1, 1, 1, 1D +

x@1D x@2D x@3D + x@1D x@2D x@4D + x@1D x@3D x@4D + x@2D x@3D x@4D

-h@1D3 h@2D2+ 2 h@1D h@2D3

- 2 h@1D4 h@3D + 6 h@1D2 h@2D h@3D - 3 h@2D2 h@3D - 3 h@1D h@3D2+

4 m@2, 2D +

6p@1D3 p@2D +

6p@1D2 p@3D +

6p@2D p@3D + p@3D2

+ x@1D x@2D x@3D

-h@7D + 4 m@2, 2D +

6p@1D3 p@2D +

6p@1D2 p@3D +

6p@2D p@3D + p@3D2

+ s@3, 1, 1, 1, 1, 1, 1, 1, 1D + x@1D x@2D x@3D

Matrices� JTH, JTE : Jacobi-Trudi Matrices

JTH@<Λ>D

Gives Jacobi-Trudi Matrix in h-functions, whose determinant equals s[Λ].

JTH@83, 2, 1, 1<D �� MatrixForm

Det@%D �� HToS

h@3D h@4D h@5D h@6Dh@1D h@2D h@3D h@4D

0 1 h@1D h@2D0 0 1 h@1D

s@3, 2, 1, 1D

JTE@<Λ>D

Gives Jacobi-Trudi Matrix in e-functions, whose determinant equals s[Λ].

10 symfun31-doc.nb

JTE@83, 2, 1, 1<D �� MatrixForm

Det@%D �� EToS

e@4D e@5D e@6De@1D e@2D e@3D

0 1 e@1Ds@3, 2, 1, 1D

� K, KInv,KStar,KTr : Kostka matrices

K@<n>D

Returns the p(n) × p(n) Kostka matrix

K@4D �� MatrixForm

1 1 1 1 1

0 1 1 2 3

0 0 1 1 2

0 0 0 1 3

0 0 0 0 1

K@<n>, <k>D

Returns the p(n, k) × p(n, k) "restricted" Kostka matrix.

K@6, 3D �� MatrixForm

1 1 1 1 1 1 1

0 1 1 2 1 2 2

0 0 1 1 1 2 3

0 0 0 1 0 1 1

0 0 0 0 1 1 1

0 0 0 0 0 1 2

0 0 0 0 0 0 1

KInv@<n>D

Returns the p(n) × p(n) inverse Kostka matrix

KInv@4D �� MatrixForm

1 -1 0 1 -1

0 1 -1 -1 2

0 0 1 -1 1

0 0 0 1 -3

0 0 0 0 1

KInv@<n>, <k>D

Returns the p(n, k) × p(n, k) inverse "restricted" Kostka matrix

KInv@6, 3D �� MatrixForm

1 -1 0 1 0 0 0

0 1 -1 -1 0 1 0

0 0 1 -1 -1 0 -1

0 0 0 1 0 -1 1

0 0 0 0 1 -1 1

0 0 0 0 0 1 -2

0 0 0 0 0 0 1

KTr@<n>D

symfun31-doc.nb 11

Returns the transpose of the p(n) × p(n) Kostka matrix

KTr@4D �� MatrixForm

1 0 0 0 0

1 1 0 0 0

1 1 1 0 0

1 2 1 1 0

1 3 2 3 1

KTr@<n>, <k>D

Returns the transpose of the p(n, k) × p(n, k) "restricted" Kostka matrix

KTr@6, 3D �� MatrixForm

1 0 0 0 0 0 0

1 1 0 0 0 0 0

1 1 1 0 0 0 0

1 2 1 1 0 0 0

1 1 1 0 1 0 0

1 2 2 1 1 1 0

1 2 3 1 1 2 1

KStar@<n>D

Returns the inverse transpose of the p(n) × p(n) Kostka matrix

KStar@4D �� MatrixForm

1 0 0 0 0

-1 1 0 0 0

0 -1 1 0 0

1 -1 -1 1 0

-1 2 1 -3 1

KStar@<n>, <k>D

Returns the inverse transpose of the p(n, k) × p(n, k) "restricted" Kostka matrix

KStar@6, 3D �� MatrixForm

1 0 0 0 0 0 0

-1 1 0 0 0 0 0

0 -1 1 0 0 0 0

1 -1 -1 1 0 0 0

0 0 -1 0 1 0 0

0 1 0 -1 -1 1 0

0 0 -1 1 1 -2 1

12 symfun31-doc.nb

� Bigger Examples

K@8D �� MatrixForm

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 1 1 2 1 2 3 1 2 2 3 4 2 3 3 4 5 3 4 5 6 7

0 0 1 1 1 2 3 1 2 3 4 6 3 4 5 7 10 6 8 11 15 20

0 0 0 1 0 1 3 0 1 1 3 6 1 3 3 6 10 3 6 10 15 21

0 0 0 0 1 1 1 1 2 2 3 4 3 4 5 7 10 6 9 13 19 28

0 0 0 0 0 1 2 0 1 2 4 8 2 4 6 11 20 8 14 24 40 64

0 0 0 0 0 0 1 0 0 0 1 4 0 1 1 4 10 1 4 10 20 35

0 0 0 0 0 0 0 1 1 1 1 1 1 2 2 3 4 3 4 6 9 14

0 0 0 0 0 0 0 0 1 1 2 3 2 4 5 9 15 7 13 23 40 70

0 0 0 0 0 0 0 0 0 1 1 2 1 1 3 5 10 6 9 16 30 56

0 0 0 0 0 0 0 0 0 0 1 3 0 1 2 6 15 3 9 21 45 90

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 5 0 1 5 15 35

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 3 6 11 21 42

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 6 2 5 12 26 56

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 5 3 6 13 30 70

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 0 2 8 24 64

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 6 21

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 5 14

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 9 28

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 20

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

KInv@24, 4D �� MatrixForm �� Timing

KTr@12, 5D

KStar@6D �� MatrixForm

1 0 0 0 0 0 0 0 0 0 0

-1 1 0 0 0 0 0 0 0 0 0

0 -1 1 0 0 0 0 0 0 0 0

1 -1 -1 1 0 0 0 0 0 0 0

0 0 -1 0 1 0 0 0 0 0 0

0 1 0 -1 -1 1 0 0 0 0 0

-1 1 1 -1 1 -2 1 0 0 0 0

0 0 -1 1 1 -2 0 1 0 0 0

0 -1 1 1 0 0 -1 -1 1 0 0

1 -1 -2 1 -1 4 -1 1 -3 1 0

-1 2 2 -3 1 -6 4 -1 6 -5 1

� Chi, ChiInv : Character matrices

Chi@<n>D, Chi2@<n>D

Returns the p(n) × p(n) character matrix.

Chi@4D �� MatrixForm

1 1 1 1 1

-1 0 -1 1 3

0 -1 2 0 2

1 0 -1 -1 3

-1 1 1 -1 1

ChiInv@<n>D, ChiInv2@<n>D

symfun31-doc.nb 13

Returns the p(n) × p(n) inverse of the character matrix.

ChiInv@4D �� MatrixForm

� Bigger Examples

Chi@7D �� MatrixForm �� Timing

:0.094 Second,

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

-1 0 -1 1 -1 0 2 0 -1 1 3 0 2 4 6

0 -1 1 -1 0 0 0 -1 2 0 2 2 2 6 14

1 0 0 0 1 -1 1 0 -1 -1 3 -3 -1 5 15

0 0 -1 -1 1 0 -2 2 -1 1 -1 0 2 4 14

0 1 0 0 -1 1 -1 -1 -1 -1 -1 1 -1 5 35

-1 0 0 0 0 0 0 2 2 0 2 0 -4 0 20

0 0 1 1 -1 -1 -1 0 1 1 -3 -3 1 1 21

0 0 -1 1 1 -1 1 0 1 -1 -3 3 1 -1 21

0 -1 0 0 1 1 1 -1 -1 1 -1 -1 -1 -5 35

1 0 0 0 -1 -1 -1 0 -1 1 3 3 -1 -5 15

0 0 1 -1 -1 0 2 2 -1 -1 -1 0 2 -4 14

0 1 -1 -1 0 0 0 -1 2 0 2 -2 2 -6 14

-1 0 1 1 1 0 -2 0 -1 -1 3 0 2 -4 6

1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1

Chi2@7D �� MatrixForm �� Timing

:21.828 Second,

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

-1 0 -1 1 -1 0 2 0 -1 1 3 0 2 4 6

0 -1 1 -1 0 0 0 -1 2 0 2 2 2 6 14

1 0 0 0 1 -1 1 0 -1 -1 3 -3 -1 5 15

0 0 -1 -1 1 0 -2 2 -1 1 -1 0 2 4 14

0 1 0 0 -1 1 -1 -1 -1 -1 -1 1 -1 5 35

-1 0 0 0 0 0 0 2 2 0 2 0 -4 0 20

0 0 1 1 -1 -1 -1 0 1 1 -3 -3 1 1 21

0 0 -1 1 1 -1 1 0 1 -1 -3 3 1 -1 21

0 -1 0 0 1 1 1 -1 -1 1 -1 -1 -1 -5 35

1 0 0 0 -1 -1 -1 0 -1 1 3 3 -1 -5 15

0 0 1 -1 -1 0 2 2 -1 -1 -1 0 2 -4 14

0 1 -1 -1 0 0 0 -1 2 0 2 -2 2 -6 14

-1 0 1 1 1 0 -2 0 -1 -1 3 0 2 -4 6

1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1

chi10 = Chi@10D

Det@chi10DApply@Times, Flatten@Partitions@10DDD9 263 795 272 057 860 957 392 207 640 004 657 152 000 000 000

9 263 795 272 057 860 957 392 207 640 004 657 152 000 000 000

This last result illustrates the remarkable result that the determinant of the character matrix of Sn equals theproduct of all of the parts in all of the partitions of n.

14 symfun31-doc.nb

This last result illustrates the remarkable result that the determinant of the character matrix of Sn equals theproduct of all of the parts in all of the partitions of n.

� J (transposition matrix), SToEMat, EToSMat, HToH2Mat, H2ToHMat (special matrices for restricted variable situations)

J@<n>D

Returns the p(n) × p(n) "transposition" matrix that takes every partition to its conjugate

J@6D �� MatrixForm

0 0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0

SToEMat@<n>, <k>D

Returns the p(n, k) × p(n, k) transition matrix from the S basis to the E basis in a situation where the number ofvariables is restricted to k

SToEMat@8, 3D �� MatrixForm

1 -7 15 6 -10 -20 1 12 6 -3

0 1 -5 -1 6 8 -1 -9 -3 3

0 0 1 -1 -3 2 1 3 -2 -1

0 0 0 1 0 -4 0 3 3 -2

0 0 0 0 1 -2 -1 1 2 -1

0 0 0 0 0 1 0 -2 -1 2

0 0 0 0 0 0 1 -3 1 2

0 0 0 0 0 0 0 1 -1 -1

0 0 0 0 0 0 0 0 1 -1

0 0 0 0 0 0 0 0 0 1

EToSMat@<n>, <k>D

Returns the p(n, k) × p(n, k) transition matrix from the E basis to the S basis in a situation where the number ofvariables is restricted to k

EToSMat@8, 3D �� MatrixForm

1 7 20 21 28 64 14 70 56 42

0 1 5 6 9 24 5 30 26 21

0 0 1 1 3 8 2 13 12 11

0 0 0 1 0 4 0 5 6 5

0 0 0 0 1 2 1 6 5 6

0 0 0 0 0 1 0 2 3 3

0 0 0 0 0 0 1 3 2 3

0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 1 1

0 0 0 0 0 0 0 0 0 1

HToH2Mat@<n>, <k>D

Returns a p(n, k) × p(n, k) matrix that converts from the basis of H-restricted expressed by HList[n,k] to the basisof H-restricted expressed by HList2[n,k]

symfun31-doc.nb 15

Returns a p(n, k) × p(n, k) matrix that converts from the basis of H-restricted expressed by HList[n,k] to the basisof H-restricted expressed by HList2[n,k]

HRestrictionRules@5, 2DHToH2Mat@5, 2D �� MatrixForm

H2ToHMat@5, 2D �� MatrixForm

9h@5D ® -2 h@1D3 h@2D + 3 h@1D h@2D2, h@1D h@4D ® -h@1D5+ h@1D3 h@2D + h@1D h@2D2,

h@2D h@3D ® -h@1D3 h@2D + 2 h@1D h@2D2, h@1D2 h@3D ® -h@1D5+ 2 h@1D3 h@2D,

h@1D h@2D2® h@1D h@2D2, h@1D3 h@2D ® h@1D3 h@2D, h@1D5

® h@1D5=

3 -2 0

1 1 -1

2 -1 0

-1 0 2

-2 0 3

-3 -1 5

� Bigger examples

SToEMat@15, 3D �� MatrixForm

1 -14 78 13 -220 -132 330 495 55 -252 -840 -360 84 630 756 84 -8 -168 -

0 1 -12 -1 55 22 -120 -135 -10 126 336 108 -56 -350 -336 -28 7 126

0 0 1 -1 -10 9 36 -18 -9 -56 -28 48 35 105 -42 -21 -6 -69

0 0 0 1 0 -11 0 45 10 0 -84 -72 0 70 168 28 0 -21 -

0 0 0 0 1 -2 -8 15 2 21 -28 -29 -20 -5 90 13 5 30

0 0 0 0 0 1 0 -9 -1 0 28 16 0 -35 -63 -7 0 15

0 0 0 0 0 0 1 -3 1 -6 17 -2 10 -20 -27 4 -4 -6

0 0 0 0 0 0 0 1 -1 0 -7 6 0 15 -3 -6 0 -10

0 0 0 0 0 0 0 0 1 0 0 -8 0 0 21 7 0 0

0 0 0 0 0 0 0 0 0 1 -4 3 -4 15 -6 -3 3 -6

0 0 0 0 0 0 0 0 0 0 1 -2 0 -5 9 2 0 6

0 0 0 0 0 0 0 0 0 0 0 1 0 0 -6 -1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 -5 6 -1 -2 9

0 0 0 0 0 0 0 0 0 0 0 0 0 1 -3 1 0 -3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

� Other matrices: PEMat, PHMat, EPMat, EHMat, HPMat, HEMat

PEMat@<n>D

Gives n × n matrix whose determinant is p[n] in terms of e[1],...,e[n]

16 symfun31-doc.nb

PEMat@5D �� MatrixForm

Det@%DEToP@%D

e@1D 1 0 0 0

2 e@2D e@1D 1 0 0

3 e@3D e@2D e@1D 1 0

4 e@4D e@3D e@2D e@1D 1

5 e@5D e@4D e@3D e@2D e@1D

e@1D5- 5 e@1D3 e@2D + 5 e@1D e@2D2

+ 5 e@1D2 e@3D - 5 e@2D e@3D - 5 e@1D e@4D + 5 e@5Dp@5D

PHMat@<n>D

Gives n × n matrix whose determinant is ±p[n] in terms of h[1],...,h[n] (The ± is implemented in the ph function.)

PHMat@5D �� MatrixForm

Det@%DHToP@%D

h@1D 1 0 0 0

2 h@2D h@1D 1 0 0

3 h@3D h@2D h@1D 1 0

4 h@4D h@3D h@2D h@1D 1

5 h@5D h@4D h@3D h@2D h@1D

h@1D5- 5 h@1D3 h@2D + 5 h@1D h@2D2

+ 5 h@1D2 h@3D - 5 h@2D h@3D - 5 h@1D h@4D + 5 h@5Dp@5D

EPMat@<n>D

Gives n × n matrix whose determinant is n!*e[n] in terms of p[1],...,p[n]

EPMat@5D �� MatrixForm

Det@%DPToE@%D

p@1D 1 0 0 0

p@2D p@1D 2 0 0

p@3D p@2D p@1D 3 0

p@4D p@3D p@2D p@1D 4

p@5D p@4D p@3D p@2D p@1D

p@1D5- 10 p@1D3 p@2D + 15 p@1D p@2D2

+ 20 p@1D2 p@3D - 20 p@2D p@3D - 30 p@1D p@4D + 24 p@5D120 e@5D

EHMat@<n>D

Gives n × n matrix whose determinant is e[n] in terms of h[1],...,h[n]

symfun31-doc.nb 17

EHMat@5D �� MatrixForm

Det@%DHToE@%D

h@1D h@2D h@3D h@4D h@5D1 h@1D h@2D h@3D h@4D0 1 h@1D h@2D h@3D0 0 1 h@1D h@2D0 0 0 1 h@1D

h@1D5- 4 h@1D3 h@2D + 3 h@1D h@2D2

+ 3 h@1D2 h@3D - 2 h@2D h@3D - 2 h@1D h@4D + h@5De@5D

HPMat@<n>D

Gives n × n matrix whose determinant is n!*h[n] in terms of p[1],...,p[n]

HPMat@5D �� MatrixForm

Det@%DPToH@%D

p@1D -1 0 0 0

p@2D p@1D -2 0 0

p@3D p@2D p@1D -3 0

p@4D p@3D p@2D p@1D -4

p@5D p@4D p@3D p@2D p@1D

p@1D5+ 10 p@1D3 p@2D + 15 p@1D p@2D2

+ 20 p@1D2 p@3D + 20 p@2D p@3D + 30 p@1D p@4D + 24 p@5D120 h@5D

HEMat@<n>D

Gives n × n matrix whose determinant is h[n] in terms of e[1],...,e[n]

HEMat@5D �� MatrixForm

Det@%DEToH@%D

e@1D e@2D e@3D e@4D e@5D1 e@1D e@2D e@3D e@4D0 1 e@1D e@2D e@3D0 0 1 e@1D e@2D0 0 0 1 e@1D

e@1D5- 4 e@1D3 e@2D + 3 e@1D e@2D2

+ 3 e@1D2 e@3D - 2 e@2D e@3D - 2 e@1D e@4D + e@5Dh@5D

Basis Conversions� XTo[HEMPSAll]

� XToM

XToM@<f>D

If f is a symmetric function expressed in terms of x[1],x[2],..., returns f expressed in terms of the m[lambda]'s.(Assumes that f is symmetric but does not test for this.)

18 symfun31-doc.nb

XToM@x@1D^3 + 3 x@1D x@2D x@3D + x@3D^3 + x@2D^3D

m@3D + 3 m@1, 1, 1D

NumberOfVars@<f>D

Returns number of x[i] variables in an expression involving x[i]'s.

NumberOfVars@x@1D x@2D + x@3DD

NumberOfVars@x@1D^2 + x@2D^2D

XToM@x@1D^2 + x@2D^2D

XToM@Hx@1D + x@2DL Hx@1D + x@3DL Hx@2D + x@3DLD

m@2, 1D + 2 m@1, 1, 1D

� XTo[Others]

XToB@<f>D

Converts f (expressed in terms of x[i]) into basis B, where B can be H,E,P,S, or "All".

f = Product@x@iD + x@jD, 8i, 1, 3<, 8j, i + 1, 3<Df �� XToH

f �� XToE

f �� XToP

f �� XToS

f �� XToM

Hx@1D + x@2DL Hx@1D + x@3DL Hx@2D + x@3DL

h@1D h@2D - h@3D

e@1D e@2D - e@3D

s@2, 1D

m@2, 1D + 2 m@1, 1, 1D

� STo[HEMPXAll]

� SToH

SToH@<f>,< opts>DOptions@SToHD=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of s functions, SToH returns the same functionexpressed as a linear combination of the appropriate h functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

symfun31-doc.nb 19

SToH@s@4D + 3 s@2, 2DD% �� HToS

3 h@2D2- 3 h@1D h@3D + h@4D

s@4D + 3 s@2, 2D

SToH@s@4D + 3 s@2, 2D, Vars -> 2D

2 h@1D4- 5 h@1D2 h@2D + 4 h@2D2

HToX@% - %%, 2D

� Bigger Examples

� SToE

SToE@<f>, <opts>DOptions@SToED=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of s functions, SToE returns the same functionexpressed as a linear combination of the appropriate e functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

SToE@4 s@4D - 3 s@3, 1D + 2 s@2, 2D - s@1, 1, 1, 1DD% �� EToS

4 e@1D4- 15 e@1D2 e@2D + 9 e@2D2

+ 9 e@1D e@3D - 8 e@4D4 s@4D + 2 s@2, 2D - 3 s@3, 1D - s@1, 1, 1, 1D

SToE@4 s@4D - 3 s@3, 1D + 2 s@2, 2D - s@1, 1, 1, 1D, Vars ® 2D

4 e@1D4- 15 e@1D2 e@2D + 9 e@2D2

EToX@% - %%, 2D

� Bigger examples

� SToM

SToM@<f>, <opts>DOptions@SToMD=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of s functions, SToM returns the same functionexpressed as a linear combination of the appropriate m functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

SToM@3 s@5D - 9 s@4, 1D + s@1, 1, 1, 1, 1DD% �� MToS

3 m@5D - 6 m@3, 2D - 6 m@4, 1D - 15 m@2, 2, 1D - 15 m@3, 1, 1D - 24 m@2, 1, 1, 1D - 32 m@1, 1, 1, 1, 1D

3 s@5D - 9 s@4, 1D + s@1, 1, 1, 1, 1D

20 symfun31-doc.nb

SToM@s@3, 2, 1DD% �� MToS

2 m@2, 2, 2D + m@3, 2, 1D + 4 m@2, 2, 1, 1D +

2 m@3, 1, 1, 1D + 8 m@2, 1, 1, 1, 1D + 16 m@1, 1, 1, 1, 1, 1Ds@3, 2, 1D

SToM@s@3, 2, 1D, Vars ® 4D

2 m@2, 2, 2D + m@3, 2, 1D + 4 m@2, 2, 1, 1D + 2 m@3, 1, 1, 1D

� Bigger examples

� SToP

SToP@<f>,<opts>DOptions@SToPD=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of s functions, SToP returns the same functionexpressed as a linear combination of the appropriate p functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

SToP@2 s@1D + s@3D + 2 s@2, 1D + s@1, 1, 1DD% �� PToS

2 p@1D + p@1D3

2 s@1D + s@3D + 2 s@2, 1D + s@1, 1, 1D

SToP@s@4, 3, 2, 1DD% �� PToS

p@1D10

315p@1D7 p@3D -

27p@1D p@3D3

75p@1D5 p@5D +

15p@1D2 p@3D p@5D -

21p@1D3 p@7D +

21p@3D p@7D

s@4, 3, 2, 1D

SToP@s@4, 3, 2, 1D, Vars ® 4D

p@1D10

288p@1D6 p@2D2

24p@1D4 p@2D3

64p@1D2 p@2D4

144p@1D7 p@3D +

48p@1D5 p@2D p@3D -

144p@1D3 p@2D2 p@3D +

48p@1D p@2D3 p@3D -

72p@1D4 p@3D2

12p@1D2 p@2D p@3D2

72p@2D2 p@3D2

27p@1D p@3D3

144p@1D6 p@4D -

12p@1D4 p@2D p@4D +

16p@1D2 p@2D2 p@4D +

72p@1D3 p@3D p@4D -

24p@1D p@2D p@3D p@4D +

36p@3D2 p@4D -

16p@1D2 p@4D2

symfun31-doc.nb 21

PToS@%DPToS@%%, Vars ® 4Ds@4, 3, 2, 1D - s@2, 2, 2, 2, 2D - s@3, 2, 2, 2, 1D + s@3, 3, 2, 1, 1D +

s@4, 2, 2, 1, 1D + s@4, 3, 1, 1, 1D - 2 s@2, 2, 2, 2, 1, 1D + s@3, 3, 1, 1, 1, 1D +

s@4, 2, 1, 1, 1, 1D - 2 s@2, 2, 2, 1, 1, 1, 1D - 2 s@2, 2, 1, 1, 1, 1, 1, 1D -

s@3, 1, 1, 1, 1, 1, 1, 1D - 2 s@2, 1, 1, 1, 1, 1, 1, 1, 1D - s@1, 1, 1, 1, 1, 1, 1, 1, 1, 1Ds@4, 3, 2, 1D

� SToX

SToX@<f>,<vars>D

If f is a symmetric function expressed as a linear combination of s functions, SToX returns the same functionexpressed in variables x[1], ..., x[n].

SToX@2 s@1D + s@3D + 2 s@2, 1D + s@1, 1, 1D, 4DXToS@%D

2 x@1D + x@1D3+ 2 x@2D + 3 x@1D2 x@2D + 3 x@1D x@2D2

+ x@2D3+ 2 x@3D +

3 x@1D2 x@3D + 6 x@1D x@2D x@3D + 3 x@2D2 x@3D + 3 x@1D x@3D2+ 3 x@2D x@3D2

x@3D3+ 2 x@4D + 3 x@1D2 x@4D + 6 x@1D x@2D x@4D + 3 x@2D2 x@4D + 6 x@1D x@3D x@4D +

6 x@2D x@3D x@4D + 3 x@3D2 x@4D + 3 x@1D x@4D2+ 3 x@2D x@4D2

+ 3 x@3D x@4D2+ x@4D3

2 s@1D + s@3D + 2 s@2, 1D + s@1, 1, 1D

SToX@s@3, 2, 1D, 3D

x@1D3 x@2D2 x@3D + x@1D2 x@2D3 x@3D + x@1D3 x@2D x@3D2+

2 x@1D2 x@2D2 x@3D2+ x@1D x@2D3 x@3D2

+ x@1D2 x@2D x@3D3+ x@1D x@2D2 x@3D3

SToX@s@4, 3, 2, 1D, 4DXToS@%D

x@1D4 x@2D3 x@3D2 x@4D + x@1D3 x@2D4 x@3D2 x@4D +

x@1D4 x@2D2 x@3D3 x@4D + 2 x@1D3 x@2D3 x@3D3 x@4D + x@1D2 x@2D4 x@3D3 x@4D +

x@1D3 x@2D2 x@3D4 x@4D + x@1D2 x@2D3 x@3D4 x@4D + x@1D4 x@2D3 x@3D x@4D2+

x@1D3 x@2D4 x@3D x@4D2+ 2 x@1D4 x@2D2 x@3D2 x@4D2

+ 4 x@1D3 x@2D3 x@3D2 x@4D2+

2 x@1D2 x@2D4 x@3D2 x@4D2+ x@1D4 x@2D x@3D3 x@4D2

+ 4 x@1D3 x@2D2 x@3D3 x@4D2+

4 x@1D2 x@2D3 x@3D3 x@4D2+ x@1D x@2D4 x@3D3 x@4D2

+ x@1D3 x@2D x@3D4 x@4D2+

2 x@1D2 x@2D2 x@3D4 x@4D2+ x@1D x@2D3 x@3D4 x@4D2

+ x@1D4 x@2D2 x@3D x@4D3+

2 x@1D3 x@2D3 x@3D x@4D3+ x@1D2 x@2D4 x@3D x@4D3

+ x@1D4 x@2D x@3D2 x@4D3+

4 x@1D3 x@2D2 x@3D2 x@4D3+ 4 x@1D2 x@2D3 x@3D2 x@4D3

+ x@1D x@2D4 x@3D2 x@4D3+

2 x@1D3 x@2D x@3D3 x@4D3+ 4 x@1D2 x@2D2 x@3D3 x@4D3

+ 2 x@1D x@2D3 x@3D3 x@4D3+

x@1D2 x@2D x@3D4 x@4D3+ x@1D x@2D2 x@3D4 x@4D3

+ x@1D3 x@2D2 x@3D x@4D4+

x@1D2 x@2D3 x@3D x@4D4+ x@1D3 x@2D x@3D2 x@4D4

+ 2 x@1D2 x@2D2 x@3D2 x@4D4+

x@1D x@2D3 x@3D2 x@4D4+ x@1D2 x@2D x@3D3 x@4D4

+ x@1D x@2D2 x@3D3 x@4D4

s@4, 3, 2, 1D

� SToAll

SToAll@<f>, <opts>DOptions@SToAllD=8Targets®8E,H,P,M, X<, Vars®0<

If f is a symmetric function expressed as a linear combination of s functions, SToAll returns the same functionexpressed as linear combinations of the appropriate functions as specified in Targets. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer. Note that the X's will only be displayed if Vars is specified.

22 symfun31-doc.nb

SToAll@s@3, 2, 1DD �� ColumnForm

s@3, 2, 1De@1D e@2D e@3D - e@3D2

- e@1D2 e@4D + e@1D e@5Dh@1D h@2D h@3D - h@3D2

- h@1D2 h@4D + h@1D h@5Dp@1D6

9p@1D3 p@3D -

5p@1D p@5D

2 m@2, 2, 2D + m@3, 2, 1D + 4 m@2, 2, 1, 1D + 2 m@3, 1, 1, 1D + 8 m@2, 1, 1, 1, 1D + 16 m@1, 1, 1, 1, 1,

SToAll@s@3, 2, 1D - 2 s@2, 2, 2D, Vars ® 3, Targets ® 8E, M, X<D

9-2 s@2, 2, 2D + s@3, 2, 1D, e@1D e@2D e@3D - 3 e@3D2, m@3, 2, 1D, x@1D3 x@2D2 x@3D +

x@1D2 x@2D3 x@3D + x@1D3 x@2D x@3D2+ x@1D x@2D3 x@3D2

+ x@1D2 x@2D x@3D3+ x@1D x@2D2 x@3D3=

� PTo[HEMSXAll]

� PRestrictionRules

PRestrictionRules@<deg>, <var>D

PRestrictionRules[deg, var] creates a list of rules that express p functions of degree deg in terms of other pfuntions of degree deg when the number of variables is restricted to var. The basis used for this is p[lambda]functions where lambda[[1]]£var (note that this is NOT the same basis generated by PList[deg, var]).

PRestrictionRules@4, 2DPList@4, 2D

:p@4D ® -

2p@1D4

+ p@1D2 p@2D +

2, p@1D p@3D ® -

2p@1D4

2p@1D2 p@2D,

p@2D2® p@2D2, p@1D2 p@2D ® p@1D2 p@2D, p@1D4

® p@1D4>

9p@4D, p@1D p@3D, p@2D2=

� PToH

PToH@<f>,< opts>DOptions@PToHD=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of p functions, PToH returns the same functionexpressed as a linear combination of the appropriate h functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

PToH@p@1D^2 p@4DD% �� HToP

-h@1D6+ 4 h@1D4 h@2D - 2 h@1D2 h@2D2

- 4 h@1D3 h@3D + 4 h@1D2 h@4D

p@1D2 p@4D

Restrict@%, 3DPToH@p@1D^2 p@4D, Vars ® 3D

3 h@1D6- 8 h@1D4 h@2D + 2 h@1D2 h@2D2

+ 4 h@1D3 h@3D

3 h@1D6- 8 h@1D4 h@2D + 2 h@1D2 h@2D2

+ 4 h@1D3 h@3D

symfun31-doc.nb 23

Restrict@p@1D^2 p@4D, 3DPToH@%D

6- p@1D4 p@2D +

2p@1D2 p@2D2

3p@1D3 p@3D

3 h@1D6- 8 h@1D4 h@2D + 2 h@1D2 h@2D2

+ 4 h@1D3 h@3D

� PToE

PToE@<f>,< opts>DOptions@PToED=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of p functions, PToE returns the same functionexpressed as a linear combination of the appropriate e functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

PToE@p@1D^2 p@4DD% �� EToP

e@1D6- 4 e@1D4 e@2D + 2 e@1D2 e@2D2

+ 4 e@1D3 e@3D - 4 e@1D2 e@4D

p@1D2 p@4D

Restrict@%, 3DPToE@p@1D^2 p@4D, Vars ® 3D

e@1D6- 4 e@1D4 e@2D + 2 e@1D2 e@2D2

+ 4 e@1D3 e@3D

e@1D6- 4 e@1D4 e@2D + 2 e@1D2 e@2D2

+ 4 e@1D3 e@3D

� PToM

PToM@<f>,< opts>DOptions@PToMD=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of p functions, PToM returns the same functionexpressed as a linear combination of the appropriate m functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

PToM@p@3D - 5 p@1D^3D% �� MToP

-4 m@3D - 15 m@2, 1D - 30 m@1, 1, 1D

-5 p@1D3+ p@3D

Restrict@%, 2DPToM@p@3D - 5 p@1D^3, Vars ® 2D

2p@1D3

2p@1D p@2D

-4 m@3D - 15 m@2, 1D

24 symfun31-doc.nb

PToM@p@4D - 3 p@3D p@1D + 2 p@1D^4DRestrict@%, 3DPToM@p@4D - 3 p@3D p@1D + 2 p@1D^4, Vars ® 3D12 m@2, 2D + 5 m@3, 1D + 24 m@2, 1, 1D + 48 m@1, 1, 1, 1D

12 m@2, 2D + 5 m@3, 1D + 24 m@2, 1, 1D

� PToS

PToS@<f>,<options>DOptions@PToSD=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of p functions, PToS returns the same functionexpressed as a linear combination of the appropriate s functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

PToS@p@1D^3 + 2 p@3DD% �� SToP

3 s@3D + 3 s@1, 1, 1D

p@1D3+ 2 p@3D

PToS@p@1D^3 + 2 p@3D, Vars ® 2D

3 s@3D

� PToX

PToX@<f>,<n>D

If f is a symmetric function expressed as a linear combination of p functions, PToX returns the same functionexpressed in variables x[1], ..., x[n].

PToX@p@2D p@1D^2, 3D% �� XToP

x@1D4+ 2 x@1D3 x@2D + 2 x@1D2 x@2D2

+ 2 x@1D x@2D3+ x@2D4

+ 2 x@1D3 x@3D +

2 x@1D2 x@2D x@3D + 2 x@1D x@2D2 x@3D + 2 x@2D3 x@3D + 2 x@1D2 x@3D2+

2 x@1D x@2D x@3D2+ 2 x@2D2 x@3D2

+ 2 x@1D x@3D3+ 2 x@2D x@3D3

+ x@3D4

p@1D2 p@2D

� PToAll

PToAll@<f>, <opts>DOptions@PToAllD=8Targets®8E,H,S,M, X<, Vars®0<

If f is a symmetric function expressed as a linear combination of p functions, PToAll returns the same functionexpressed as linear combinations of the appropriate functions as specified in Targets. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer. Note that the X's will only be displayed if Vars is specified.

symfun31-doc.nb 25

Sum@p@iD p@jD, 8i, 3<, 8j, 3<D;

PToAll@%D

9p@1D2+ 2 p@1D p@2D + p@2D2

+ 2 p@1D p@3D + 2 p@2D p@3D + p@3D2,

e@1D2+ 2 e@1D3

+ 3 e@1D4+ 2 e@1D5

+ e@1D6- 4 e@1D e@2D - 10 e@1D2 e@2D -

10 e@1D3 e@2D - 6 e@1D4 e@2D + 4 e@2D2+ 12 e@1D e@2D2

+ 9 e@1D2 e@2D2+ 6 e@1D e@3D +

6 e@1D2 e@3D + 6 e@1D3 e@3D - 12 e@2D e@3D - 18 e@1D e@2D e@3D + 9 e@3D2,

h@1D2- 2 h@1D3

+ 3 h@1D4- 2 h@1D5

+ h@1D6+ 4 h@1D h@2D - 10 h@1D2 h@2D +

10 h@1D3 h@2D - 6 h@1D4 h@2D + 4 h@2D2- 12 h@1D h@2D2

+ 9 h@1D2 h@2D2+ 6 h@1D h@3D -

6 h@1D2 h@3D + 6 h@1D3 h@3D + 12 h@2D h@3D - 18 h@1D h@2D h@3D + 9 h@3D2,

s@2D + 2 s@3D + 3 s@4D + 2 s@5D + s@6D + s@1, 1D - s@3, 1D + 2 s@3, 2D + 2 s@3, 3D -

2 s@4, 1D - s@5, 1D - 2 s@1, 1, 1D - s@2, 1, 1D - 2 s@2, 2, 1D + 2 s@2, 2, 2D -

2 s@3, 2, 1D + s@4, 1, 1D + 3 s@1, 1, 1, 1D + 2 s@2, 1, 1, 1D + s@3, 1, 1, 1D -

2 s@1, 1, 1, 1, 1D - s@2, 1, 1, 1, 1D + s@1, 1, 1, 1, 1, 1D, m@2D + 2 m@3D + 3 m@4D +

2 m@5D + m@6D + 2 m@1, 1D + 2 m@2, 1D + 2 m@2, 2D + 2 m@3, 1D + 2 m@3, 2D + 2 m@3, 3D=

Sum@p@iD p@jD, 8i, 3<, 8j, 3<D;

PToAll@%, Vars ® 2, Targets ® 8E, M, X<D

9p@1D2+ 2 p@1D p@2D + p@2D2

+ 2 p@1D p@3D + 2 p@2D p@3D + p@3D2,

e@1D2+ 2 e@1D3

+ 3 e@1D4+ 2 e@1D5

+ e@1D6- 4 e@1D e@2D - 10 e@1D2 e@2D -

10 e@1D3 e@2D - 6 e@1D4 e@2D + 4 e@2D2+ 12 e@1D e@2D2

+ 9 e@1D2 e@2D2,

m@2D + 2 m@3D + 3 m@4D + 2 m@5D + m@6D + 2 m@1, 1D + 2 m@2, 1D + 2 m@2, 2D + 2 m@3, 1D +

2 m@3, 2D + 2 m@3, 3D, x@1D2+ 2 x@1D3

+ 3 x@1D4+ 2 x@1D5

+ x@1D6+ 2 x@1D x@2D +

2 x@1D2 x@2D + 2 x@1D3 x@2D + x@2D2+ 2 x@1D x@2D2

+ 2 x@1D2 x@2D2+ 2 x@1D3 x@2D2

2 x@2D3+ 2 x@1D x@2D3

+ 2 x@1D2 x@2D3+ 2 x@1D3 x@2D3

+ 3 x@2D4+ 2 x@2D5

+ x@2D6=

� MTo[HEPSXAll]

� MToH

MToH@<f>, <opts>DOptions@MToHD=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of m functions, MToH returns the same functionexpressed as a linear combination of the appropriate h functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

MToH@m@1, 1, 1, 1D + 2 m@2, 2D - 3 m@3, 1D + 4 m@4DD% �� HToM

-7 h@1D4+ 26 h@1D2 h@2D - 7 h@2D2

- 31 h@1D h@3D + 23 h@4D4 m@4D + 2 m@2, 2D - 3 m@3, 1D + m@1, 1, 1, 1D

MToH@m@1, 1, 1, 1D + 2 m@2, 2D - 3 m@3, 1D + 4 m@4D, Vars ® 3D% �� HToM

16 h@1D4- 43 h@1D2 h@2D + 16 h@2D2

+ 15 h@1D h@3D4 m@4D + 2 m@2, 2D - 3 m@3, 1D + 24 m@1, 1, 1, 1D

� Bigger Examples

MToH@m@9, 1D + m@4, 3, 3D - 10 m@6, 3D + 14 m@13, 2DD �� Timing

26 symfun31-doc.nb

MToH@m@9, 1D + m@4, 3, 3D - 10 m@6, 3D + 14 m@13, 2D, Vars ® 5D �� Timing

936.037, 20 h@1D9- h@1D10

- 28 h@1D15- 180 h@1D7 h@2D + 11 h@1D8 h@2D + 238 h@1D13 h@2D +

540 h@1D5 h@2D2- 43 h@1D6 h@2D2

- 532 h@1D11 h@2D2- 590 h@1D3 h@2D3

+ 68 h@1D4 h@2D3-

700 h@1D9 h@2D3+ 150 h@1D h@2D4

- 34 h@1D2 h@2D4+ 4620 h@1D7 h@2D4

+ 2 h@2D5- 6902 h@1D5 h@2D5

3864 h@1D3 h@2D6- 518 h@1D h@2D7

+ 180 h@1D6 h@3D - 11 h@1D7 h@3D - 210 h@1D12 h@3D -

930 h@1D4 h@2D h@3D + 81 h@1D5 h@2D h@3D + 1008 h@1D10 h@2D h@3D + 1170 h@1D2 h@2D2 h@3D -

166 h@1D3 h@2D2 h@3D + 420 h@1D8 h@2D2 h@3D - 150 h@2D3 h@3D + 67 h@1D h@2D3 h@3D -

8638 h@1D6 h@2D3 h@3D + 14 546 h@1D4 h@2D4 h@3D - 7672 h@1D2 h@2D5 h@3D + 518 h@2D6 h@3D +

390 h@1D3 h@3D2- 40 h@1D4 h@3D2

- 476 h@1D9 h@3D2- 720 h@1D h@2D h@3D2

+ 141 h@1D2 h@2D h@3D2+

462 h@1D7 h@2D h@3D2- 33 h@2D2 h@3D2

+ 5068 h@1D5 h@2D2 h@3D2- 9912 h@1D3 h@2D3 h@3D2

4732 h@1D h@2D4 h@3D2+ 120 h@3D3

- 42 h@1D h@3D3- 182 h@1D6 h@3D3

- 1442 h@1D4 h@2D h@3D3+

2464 h@1D2 h@2D2 h@3D3- 924 h@2D3 h@3D3

+ 392 h@1D3 h@3D4- 154 h@1D h@2D h@3D4

42 h@3D5- 150 h@1D5 h@4D + 5 h@1D6 h@4D + 600 h@1D3 h@2D h@4D - 38 h@1D4 h@2D h@4D +

728 h@1D9 h@2D h@4D - 450 h@1D h@2D2 h@4D + 69 h@1D2 h@2D2 h@4D - 4914 h@1D7 h@2D2 h@4D -

10 h@2D3 h@4D + 10 920 h@1D5 h@2D3 h@4D - 8918 h@1D3 h@2D4 h@4D + 1820 h@1D h@2D5 h@4D -

450 h@1D2 h@3D h@4D + 37 h@1D3 h@3D h@4D - 364 h@1D8 h@3D h@4D + 360 h@2D h@3D h@4D -

117 h@1D h@2D h@3D h@4D + 5572 h@1D6 h@2D h@3D h@4D - 16 282 h@1D4 h@2D2 h@3D h@4D +

13 076 h@1D2 h@2D3 h@3D h@4D - 1456 h@2D4 h@3D h@4D + 45 h@3D2 h@4D - 1540 h@1D5 h@3D2 h@4D +

7896 h@1D3 h@2D h@3D2 h@4D - 5642 h@1D h@2D2 h@3D2 h@4D - 1652 h@1D2 h@3D3 h@4D +

784 h@2D h@3D3 h@4D + 90 h@1D h@4D2- 3 h@1D2 h@4D2

+ 420 h@1D7 h@4D2+ 12 h@2D h@4D2

3038 h@1D5 h@2D h@4D2+ 5180 h@1D3 h@2D2 h@4D2

- 1778 h@1D h@2D3 h@4D2+ 1890 h@1D4 h@3D h@4D2

4802 h@1D2 h@2D h@3D h@4D2+ 994 h@2D2 h@3D h@4D2

+ 1260 h@1D h@3D2 h@4D2- 588 h@1D3 h@4D3

476 h@1D h@2D h@4D3- 210 h@3D h@4D3

+ 150 h@1D4 h@5D - h@1D5 h@5D - 450 h@1D2 h@2D h@5D +

13 h@1D3 h@2D h@5D - 364 h@1D8 h@2D h@5D + 90 h@2D2 h@5D - 18 h@1D h@2D2 h@5D +

2366 h@1D6 h@2D2 h@5D - 4914 h@1D4 h@2D3 h@5D + 3458 h@1D2 h@2D4 h@5D - 364 h@2D5 h@5D +

360 h@1D h@3D h@5D - 15 h@1D2 h@3D h@5D + 21 h@2D h@3D h@5D - 1512 h@1D5 h@2D h@3D h@5D +

4956 h@1D3 h@2D2 h@3D h@5D - 3444 h@1D h@2D3 h@3D h@5D + 28 h@1D4 h@3D2 h@5D -

1344 h@1D2 h@2D h@3D2 h@5D + 686 h@2D2 h@3D2 h@5D + 420 h@1D h@3D3 h@5D - 90 h@4D h@5D +

3 h@1D h@4D h@5D - 420 h@1D6 h@4D h@5D + 2828 h@1D4 h@2D h@4D h@5D - 4158 h@1D2 h@2D2 h@4D h@5D +

784 h@2D3 h@4D h@5D - 1316 h@1D3 h@3D h@4D h@5D + 2632 h@1D h@2D h@3D h@4D h@5D -

630 h@3D2 h@4D h@5D + 868 h@1D2 h@4D2 h@5D - 266 h@2D h@4D2 h@5D - 182 h@1D3 h@2D h@5D2+

364 h@1D h@2D2 h@5D2- 182 h@1D2 h@3D h@5D2

+ 154 h@2D h@3D h@5D2- 210 h@1D h@4D h@5D2

70 h@5D3- 150 h@1D3 h@6D + h@1D4 h@6D + 360 h@1D h@2D h@6D - 15 h@1D2 h@2D h@6D + 6 h@2D2 h@6D -

270 h@3D h@6D + 24 h@1D h@3D h@6D - 12 h@4D h@6D + 90 h@1D2 h@7D + 2 h@1D3 h@7D - 90 h@2D h@7D +

12 h@1D h@2D h@7D - 21 h@3D h@7D - 90 h@1D h@8D - 9 h@1D2 h@8D + 90 h@9D + 9 h@1D h@9D=

� MToE

MToE@<f>, <opts>DOptions@MToED=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of m functions, MToE returns the same functionexpressed as a linear combination of the appropriate e functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

MToE@m@5D - 5 m@4, 1D + m@3, 2D - 2 m@3, 1, 1D + m@2, 2, 1DD% �� EToM

e@1D5- 10 e@1D3 e@2D + 21 e@1D e@2D2

+ 6 e@1D2 e@3D - 26 e@2D e@3D - 6 e@1D e@4D + 20 e@5Dm@5D + m@3, 2D - 5 m@4, 1D + m@2, 2, 1D - 2 m@3, 1, 1D

symfun31-doc.nb 27

MToE@m@5D - 5 m@4, 1D + m@3, 2D - 2 m@3, 1, 1D + m@2, 2, 1D, Vars ® 3D

e@1D5- 10 e@1D3 e@2D + 21 e@1D e@2D2

+ 6 e@1D2 e@3D - 26 e@2D e@3D

� Bigger Examples

MToE@m@15D - 3 m@6, 4, 2, 2, 1DD �� Timing

961.641 Second, e@1D15- 15 e@1D13 e@2D + 90 e@1D11 e@2D2

- 275 e@1D9 e@2D3+ 450 e@1D7 e@2D4

378 e@1D5 e@2D5+ 140 e@1D3 e@2D6

- 15 e@1D e@2D7+ 15 e@1D12 e@3D - 165 e@1D10 e@2D e@3D +

675 e@1D8 e@2D2 e@3D - 1260 e@1D6 e@2D3 e@3D + 1050 e@1D4 e@2D4 e@3D - 315 e@1D2 e@2D5 e@3D +

15 e@2D6 e@3D + 75 e@1D9 e@3D2- 540 e@1D7 e@2D e@3D2

+ 1260 e@1D5 e@2D2 e@3D2-

1050 e@1D3 e@2D3 e@3D2+ 225 e@1D e@2D4 e@3D2

+ 140 e@1D6 e@3D3- 525 e@1D4 e@2D e@3D3

450 e@1D2 e@2D2 e@3D3- 50 e@2D3 e@3D3

+ 75 e@1D3 e@3D4- 75 e@1D e@2D e@3D4

+ 3 e@3D5-

15 e@1D11 e@4D + 150 e@1D9 e@2D e@4D - 540 e@1D7 e@2D2 e@4D + 840 e@1D5 e@2D3 e@4D -

525 e@1D3 e@2D4 e@4D + 90 e@1D e@2D5 e@4D - 135 e@1D8 e@3D e@4D + 840 e@1D6 e@2D e@3D e@4D -

1575 e@1D4 e@2D2 e@3D e@4D + 900 e@1D2 e@2D3 e@3D e@4D - 75 e@2D4 e@3D e@4D -

315 e@1D5 e@3D2 e@4D + 900 e@1D3 e@2D e@3D2 e@4D - 450 e@1D e@2D2 e@3D2 e@4D -

150 e@1D2 e@3D3 e@4D + 60 e@2D e@3D3 e@4D + 60 e@1D7 e@4D2- 315 e@1D5 e@2D e@4D2

450 e@1D3 e@2D2 e@4D2- 150 e@1D e@2D3 e@4D2

+ 225 e@1D4 e@3D e@4D2- 450 e@1D2 e@2D e@3D e@4D2

90 e@2D2 e@3D e@4D2+ 90 e@1D e@3D2 e@4D2

- 50 e@1D3 e@4D3+ 60 e@1D e@2D e@4D3

15 e@3D e@4D3+ 15 e@1D10 e@5D - 135 e@1D8 e@2D e@5D + 420 e@1D6 e@2D2 e@5D -

525 e@1D4 e@2D3 e@5D + 225 e@1D2 e@2D4 e@5D - 15 e@2D5 e@5D + 120 e@1D7 e@3D e@5D -

630 e@1D5 e@2D e@3D e@5D + 900 e@1D3 e@2D2 e@3D e@5D - 300 e@1D e@2D3 e@3D e@5D +

225 e@1D4 e@3D2 e@5D - 450 e@1D2 e@2D e@3D2 e@5D + 90 e@2D2 e@3D2 e@5D + 60 e@1D e@3D3 e@5D -

105 e@1D6 e@4D e@5D + 450 e@1D4 e@2D e@4D e@5D - 453 e@1D2 e@2D2 e@4D e@5D +

66 e@2D3 e@4D e@5D - 294 e@1D3 e@3D e@4D e@5D + 348 e@1D e@2D e@3D e@4D e@5D -

36 e@3D2 e@4D e@5D + 84 e@1D2 e@4D2 e@5D - 51 e@2D e@4D2 e@5D + 45 e@1D5 e@5D2-

147 e@1D3 e@2D e@5D2+ 84 e@1D e@2D2 e@5D2

+ 81 e@1D2 e@3D e@5D2- 33 e@2D e@3D e@5D2

9 e@1D e@4D e@5D2- 25 e@5D3

- 15 e@1D9 e@6D + 120 e@1D7 e@2D e@6D - 315 e@1D5 e@2D2 e@6D +

300 e@1D3 e@2D3 e@6D - 75 e@1D e@2D4 e@6D - 105 e@1D6 e@3D e@6D + 450 e@1D4 e@2D e@3D e@6D -

441 e@1D2 e@2D2 e@3D e@6D + 42 e@2D3 e@3D e@6D - 168 e@1D3 e@3D2 e@6D + 216 e@1D e@2D e@3D2 e@6D -

42 e@3D3 e@6D + 90 e@1D5 e@4D e@6D - 306 e@1D3 e@2D e@4D e@6D + 192 e@1D e@2D2 e@4D e@6D +

216 e@1D2 e@3D e@4D e@6D - 96 e@2D e@3D e@4D e@6D - 81 e@1D e@4D2 e@6D - 66 e@1D4 e@5D e@6D +

144 e@1D2 e@2D e@5D e@6D - 21 e@2D2 e@5D e@6D - 120 e@1D e@3D e@5D e@6D + 48 e@4D e@5D e@6D +

21 e@1D3 e@6D2- 27 e@1D e@2D e@6D2

+ 69 e@3D e@6D2+ 15 e@1D8 e@7D - 105 e@1D6 e@2D e@7D +

225 e@1D4 e@2D2 e@7D - 165 e@1D2 e@2D3 e@7D + 45 e@2D4 e@7D + 90 e@1D5 e@3D e@7D -

264 e@1D3 e@2D e@3D e@7D + 108 e@1D e@2D2 e@3D e@7D + 72 e@1D2 e@3D2 e@7D +

24 e@2D e@3D2 e@7D - 102 e@1D4 e@4D e@7D + 258 e@1D2 e@2D e@4D e@7D - 147 e@2D2 e@4D e@7D -

126 e@1D e@3D e@4D e@7D + 87 e@4D2 e@7D + 78 e@1D3 e@5D e@7D - 36 e@1D e@2D e@5D e@7D -

30 e@3D e@5D e@7D - 63 e@1D2 e@6D e@7D - 24 e@2D e@6D e@7D + 42 e@1D e@7D2- 15 e@1D7 e@8D +

90 e@1D5 e@2D e@8D - 135 e@1D3 e@2D2 e@8D + 30 e@1D e@2D3 e@8D - 72 e@1D4 e@3D e@8D +

102 e@1D2 e@2D e@3D e@8D + 51 e@2D2 e@3D e@8D + 12 e@1D e@3D2 e@8D + 84 e@1D3 e@4D e@8D -

48 e@1D e@2D e@4D e@8D - 90 e@3D e@4D e@8D - 117 e@1D2 e@5D e@8D - 18 e@2D e@5D e@8D +

102 e@1D e@6D e@8D + 21 e@7D e@8D + 15 e@1D6 e@9D - 132 e@1D4 e@2D e@9D + 303 e@1D2 e@2D2 e@9D -

129 e@2D3 e@9D + 114 e@1D3 e@3D e@9D - 378 e@1D e@2D e@3D e@9D + 96 e@3D2 e@9D -

153 e@1D2 e@4D e@9D + 324 e@2D e@4D e@9D + 186 e@1D e@5D e@9D - 285 e@6D e@9D + 66 e@1D5 e@10D -

288 e@1D3 e@2D e@10D + 219 e@1D e@2D2 e@10D + 315 e@1D2 e@3D e@10D - 246 e@2D e@3D e@10D -

276 e@1D e@4D e@10D + 225 e@5D e@10D - 66 e@1D4 e@11D + 153 e@1D2 e@2D e@11D +

75 e@2D2 e@11D - 180 e@1D e@3D e@11D - 3 e@4D e@11D + 165 e@1D3 e@12D - 384 e@1D e@2D e@12D +

309 e@3D e@12D - 165 e@1D2 e@13D + 57 e@2D e@13D + 399 e@1D e@14D - 525 e@15D=

28 symfun31-doc.nb

MToE@m@15D - 3 m@6, 4, 2, 2, 1D, Vars ® 6D �� Timing

926.094 Second, e@1D15- 15 e@1D13 e@2D + 90 e@1D11 e@2D2

- 275 e@1D9 e@2D3+ 450 e@1D7 e@2D4

378 e@1D5 e@2D5+ 140 e@1D3 e@2D6

- 15 e@1D e@2D7+ 15 e@1D12 e@3D - 165 e@1D10 e@2D e@3D +

675 e@1D8 e@2D2 e@3D - 1260 e@1D6 e@2D3 e@3D + 1050 e@1D4 e@2D4 e@3D - 315 e@1D2 e@2D5 e@3D +

15 e@2D6 e@3D + 75 e@1D9 e@3D2- 540 e@1D7 e@2D e@3D2

+ 1260 e@1D5 e@2D2 e@3D2-

1050 e@1D3 e@2D3 e@3D2+ 225 e@1D e@2D4 e@3D2

+ 140 e@1D6 e@3D3- 525 e@1D4 e@2D e@3D3

450 e@1D2 e@2D2 e@3D3- 50 e@2D3 e@3D3

+ 75 e@1D3 e@3D4- 75 e@1D e@2D e@3D4

+ 3 e@3D5-

15 e@1D11 e@4D + 150 e@1D9 e@2D e@4D - 540 e@1D7 e@2D2 e@4D + 840 e@1D5 e@2D3 e@4D -

525 e@1D3 e@2D4 e@4D + 90 e@1D e@2D5 e@4D - 135 e@1D8 e@3D e@4D + 840 e@1D6 e@2D e@3D e@4D -

1575 e@1D4 e@2D2 e@3D e@4D + 900 e@1D2 e@2D3 e@3D e@4D - 75 e@2D4 e@3D e@4D -

315 e@1D5 e@3D2 e@4D + 900 e@1D3 e@2D e@3D2 e@4D - 450 e@1D e@2D2 e@3D2 e@4D -

150 e@1D2 e@3D3 e@4D + 60 e@2D e@3D3 e@4D + 60 e@1D7 e@4D2- 315 e@1D5 e@2D e@4D2

450 e@1D3 e@2D2 e@4D2- 150 e@1D e@2D3 e@4D2

+ 225 e@1D4 e@3D e@4D2- 450 e@1D2 e@2D e@3D e@4D2

90 e@2D2 e@3D e@4D2+ 90 e@1D e@3D2 e@4D2

- 50 e@1D3 e@4D3+ 60 e@1D e@2D e@4D3

15 e@3D e@4D3+ 15 e@1D10 e@5D - 135 e@1D8 e@2D e@5D + 420 e@1D6 e@2D2 e@5D -

525 e@1D4 e@2D3 e@5D + 225 e@1D2 e@2D4 e@5D - 15 e@2D5 e@5D + 120 e@1D7 e@3D e@5D -

630 e@1D5 e@2D e@3D e@5D + 900 e@1D3 e@2D2 e@3D e@5D - 300 e@1D e@2D3 e@3D e@5D +

225 e@1D4 e@3D2 e@5D - 450 e@1D2 e@2D e@3D2 e@5D + 90 e@2D2 e@3D2 e@5D + 60 e@1D e@3D3 e@5D -

105 e@1D6 e@4D e@5D + 450 e@1D4 e@2D e@4D e@5D - 453 e@1D2 e@2D2 e@4D e@5D +

66 e@2D3 e@4D e@5D - 294 e@1D3 e@3D e@4D e@5D + 348 e@1D e@2D e@3D e@4D e@5D -

36 e@3D2 e@4D e@5D + 84 e@1D2 e@4D2 e@5D - 51 e@2D e@4D2 e@5D + 45 e@1D5 e@5D2-

147 e@1D3 e@2D e@5D2+ 84 e@1D e@2D2 e@5D2

+ 81 e@1D2 e@3D e@5D2- 33 e@2D e@3D e@5D2

9 e@1D e@4D e@5D2- 25 e@5D3

- 15 e@1D9 e@6D + 120 e@1D7 e@2D e@6D - 315 e@1D5 e@2D2 e@6D +

300 e@1D3 e@2D3 e@6D - 75 e@1D e@2D4 e@6D - 105 e@1D6 e@3D e@6D + 450 e@1D4 e@2D e@3D e@6D -

441 e@1D2 e@2D2 e@3D e@6D + 42 e@2D3 e@3D e@6D - 168 e@1D3 e@3D2 e@6D +

216 e@1D e@2D e@3D2 e@6D - 42 e@3D3 e@6D + 90 e@1D5 e@4D e@6D - 306 e@1D3 e@2D e@4D e@6D +

192 e@1D e@2D2 e@4D e@6D + 216 e@1D2 e@3D e@4D e@6D - 96 e@2D e@3D e@4D e@6D -

81 e@1D e@4D2 e@6D - 66 e@1D4 e@5D e@6D + 144 e@1D2 e@2D e@5D e@6D - 21 e@2D2 e@5D e@6D -

120 e@1D e@3D e@5D e@6D + 48 e@4D e@5D e@6D + 21 e@1D3 e@6D2- 27 e@1D e@2D e@6D2

+ 69 e@3D e@6D2=

� MToP

MToP@<f>, <opts>DOptions@MToPD=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of m functions, MToP returns the same functionexpressed as a linear combination of the appropriate p functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

MToP@m@4, 2, 1D - 3 m@5, 2D + 2 m@3, 2, 2DD% �� PToM

p@2D2 p@3D + p@1D p@2D p@4D - 2 p@3D p@4D - 6 p@2D p@5D - p@1D p@6D + 7 p@7D-3 m@5, 2D + 2 m@3, 2, 2D + m@4, 2, 1D

MToP@m@4, 2, 1D - 3 m@5, 2D + 2 m@3, 2, 2D, Vars ® 3D

12p@1D5 p@2D +

3p@1D3 p@2D2

4p@1D p@2D3

36p@1D4 p@3D -

3p@1D2 p@2D p@3D -

12p@2D2 p@3D +

9p@1D p@3D2

symfun31-doc.nb 29

� MToS

MToS@<f>, <opts>DOptions@MToSD=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of m functions, MToS returns the same functionexpressed as a linear combination of the appropriate s functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

MToS@m@5, 2D - 6 m@3, 1, 1D + 5 m@4, 2, 1, 1D - 9 m@7DD% �� SToM

-9 s@7D - s@4, 3D + s@5, 2D + 9 s@6, 1D + 6 s@2, 2, 1D - 6 s@3, 1, 1D - s@3, 2, 2D +

s@3, 3, 1D - 10 s@5, 1, 1D + 6 s@2, 1, 1, 1D + s@2, 2, 2, 1D + 10 s@2, 2, 2, 2D -

5 s@3, 2, 2, 1D - 5 s@3, 3, 1, 1D + 10 s@4, 1, 1, 1D + 5 s@4, 2, 1, 1D - 18 s@1, 1, 1, 1, 1D -

s@2, 2, 1, 1, 1D - 5 s@2, 2, 2, 1, 1D - 10 s@3, 1, 1, 1, 1D + 10 s@3, 2, 1, 1, 1D -

15 s@4, 1, 1, 1, 1D + 11 s@2, 1, 1, 1, 1, 1D - 5 s@2, 2, 1, 1, 1, 1D + 15 s@3, 1, 1, 1, 1, 1D -

11 s@1, 1, 1, 1, 1, 1, 1D - 30 s@2, 1, 1, 1, 1, 1, 1D + 60 s@1, 1, 1, 1, 1, 1, 1, 1D-9 m@7D + m@5, 2D - 6 m@3, 1, 1D + 5 m@4, 2, 1, 1D

MToS@m@5, 2D - 6 m@3, 1, 1D + 5 m@4, 2, 1, 1D - 9 m@7D, Vars ® 3D

-9 s@7D - s@4, 3D + s@5, 2D + 9 s@6, 1D +

6 s@2, 2, 1D - 6 s@3, 1, 1D - s@3, 2, 2D + s@3, 3, 1D - 10 s@5, 1, 1D

� MToX

MToX@<f>, <n>D

If f is a symmetric function expressed as a linear combination of m functions, MToX returns the same functionexpressed in the variables x[1], ..., x[n].

MToX@m@5, 2D + m@4D, 3D% �� XToM

x@1D4+ x@1D5 x@2D2

+ x@2D4+ x@1D2 x@2D5

x@1D5 x@3D2+ x@2D5 x@3D2

+ x@3D4+ x@1D2 x@3D5

+ x@2D2 x@3D5

m@4D + m@5, 2D

� MToAll

MToAll@<f>, <opts>DOptions@MToAllD=8Targets®8E,P,S,H, X<, Vars®0<

If f is a symmetric function expressed as a linear combination of m functions, MToAll returns the same functionexpressed as linear combinations of the appropriate functions as specified in Targets. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer. Note that the X's will only be displayed if Vars is specified.

MToAll@m@3, 2, 1DD

9m@3, 2, 1D, e@1D e@2D e@3D - 3 e@3D2- 3 e@1D2 e@4D + 4 e@2D e@4D + 7 e@1D e@5D - 12 e@6D,

p@1D p@2D p@3D - p@3D2- p@2D p@4D - p@1D p@5D + 2 p@6D,

-2 s@2, 2, 2D + s@3, 2, 1D - 2 s@3, 1, 1, 1D + 4 s@2, 1, 1, 1, 1D - 6 s@1, 1, 1, 1, 1, 1D,

-6 h@1D6+ 34 h@1D4 h@2D - 48 h@1D2 h@2D2

+ 8 h@2D3- 30 h@1D3 h@3D +

61 h@1D h@2D h@3D - 15 h@3D2+ 21 h@1D2 h@4D - 20 h@2D h@4D - 17 h@1D h@5D + 12 h@6D=

30 symfun31-doc.nb

MToAll@m@4, 1, 1D - 2 m@3, 2, 1D, Vars ® 3, Targets ® 8E, S, X<D

9-2 m@3, 2, 1D + m@4, 1, 1D,

e@1D3 e@3D - 5 e@1D e@2D e@3D + 9 e@3D2, 5 s@2, 2, 2D - 3 s@3, 2, 1D + s@4, 1, 1D,

x@1D4 x@2D x@3D - 2 x@1D3 x@2D2 x@3D - 2 x@1D2 x@2D3 x@3D + x@1D x@2D4 x@3D - 2 x@1D3 x@2D x@3D2-

2 x@1D x@2D3 x@3D2- 2 x@1D2 x@2D x@3D3

- 2 x@1D x@2D2 x@3D3+ x@1D x@2D x@3D4=

� ETo[HMPSXAll]

� EToH

EToH@<f>, <opts>DOptions@EToHD=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of e functions, EToH returns the same functionexpressed as a linear combination of the appropriate h functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

EToH@e@3D e@2D e@1D + e@4D e@2DD% �� HToE

2 h@1D6- 7 h@1D4 h@2D + 6 h@1D2 h@2D2

- h@2D3+

3 h@1D3 h@3D - 3 h@1D h@2D h@3D - h@1D2 h@4D + h@2D h@4De@1D e@2D e@3D + e@2D e@4D

EToH@e@3D e@2D e@1D + e@4D e@2D, Vars ® 4D

2 h@1D6- 7 h@1D4 h@2D + 6 h@1D2 h@2D2

- h@2D3+

3 h@1D3 h@3D - 3 h@1D h@2D h@3D - h@1D2 h@4D + h@2D h@4DEToH@e@3D e@2D e@1D + e@4D e@2D, Vars ® 3D

h@1D6- 3 h@1D4 h@2D + 2 h@1D2 h@2D2

+ h@1D3 h@3D - h@1D h@2D h@3D

Product@1 + e@iD, 8i, 1, 3<D �� Expand

EToH@%D% �� HToE

1 + e@1D + e@2D + e@1D e@2D + e@3D + e@1D e@3D + e@2D e@3D + e@1D e@2D e@3D

1 + h@1D + h@1D2+ 2 h@1D3

+ h@1D4+ h@1D5

+ h@1D6- h@2D - 3 h@1D h@2D -

2 h@1D2 h@2D - 3 h@1D3 h@2D - 3 h@1D4 h@2D + 2 h@1D h@2D2+ 2 h@1D2 h@2D2

h@3D + h@1D h@3D + h@1D2 h@3D + h@1D3 h@3D - h@2D h@3D - h@1D h@2D h@3D1 + e@1D + e@2D + e@1D e@2D + e@3D + e@1D e@3D + e@2D e@3D + e@1D e@2D e@3D

� EToM

EToM@<f>, <opts>DOptions@EToMD=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of e functions, EToM returns the same functionexpressed as a linear combination of the appropriate m functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

symfun31-doc.nb 31

Product@1 + e@iD, 8i, 3<D �� Expand

EToM@%D% �� MToE

1 + e@1D + e@2D + e@1D e@2D + e@3D + e@1D e@3D + e@2D e@3D + e@1D e@2D e@3D

1 + m@1D + m@1, 1D + m@2, 1D + 4 m@1, 1, 1D + m@2, 1, 1D + m@2, 2, 1D +

3 m@2, 2, 2D + m@3, 2, 1D + 4 m@1, 1, 1, 1D + 3 m@2, 1, 1, 1D + 8 m@2, 2, 1, 1D +

3 m@3, 1, 1, 1D + 10 m@1, 1, 1, 1, 1D + 22 m@2, 1, 1, 1, 1D + 60 m@1, 1, 1, 1, 1, 1D1 + e@1D + e@2D + e@1D e@2D + e@3D + e@1D e@3D + e@2D e@3D + e@1D e@2D e@3D

Product@1 + e@iD, 8i, 3<D �� Expand

EToM@%, Vars ® 3D1 + e@1D + e@2D + e@1D e@2D + e@3D + e@1D e@3D + e@2D e@3D + e@1D e@2D e@3D

1 + m@1D + m@1, 1D + m@2, 1D + 4 m@1, 1, 1D + m@2, 1, 1D + m@2, 2, 1D + 3 m@2, 2, 2D + m@3, 2, 1D

� EToP

EToP@<f>, <opts>DOptions@EToPD=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of e functions, EToP returns the same functionexpressed as a linear combination of the appropriate p functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

EToP@e@3D e@2D e@1D - 2 e@2D^2 e@1D^2D% �� PToE

12p@1D6

3p@1D4 p@2D -

4p@1D2 p@2D2

6p@1D3 p@3D -

6p@1D p@2D p@3D

-2 e@1D2 e@2D2+ e@1D e@2D e@3D

EToP@e@3D e@2D e@1D - 2 e@2D^2 e@1D^2, Vars ® 3D

12p@1D6

3p@1D4 p@2D -

4p@1D2 p@2D2

6p@1D3 p@3D -

6p@1D p@2D p@3D

EToP@e@3D e@2D e@1D - 2 e@2D^2 e@1D^2, Vars ® 2D

2p@1D6

+ p@1D4 p@2D -

2p@1D2 p@2D2

� EToS

EToS@<f>, <opts>DOptions@EToSD=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of e functions, EToS returns the same functionexpressed as a linear combination of the appropriate s functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

32 symfun31-doc.nb

Sum@e@iD e@jD, 8i, 1, 3<, 8j, 1, 3<DEToS@%D% �� SToE

e@1D2+ 2 e@1D e@2D + e@2D2

+ 2 e@1D e@3D + 2 e@2D e@3D + e@3D2

s@2D + s@1, 1D + 2 s@2, 1D + s@2, 2D + 2 s@1, 1, 1D +

3 s@2, 1, 1D + 2 s@2, 2, 1D + s@2, 2, 2D + 3 s@1, 1, 1, 1D + 2 s@2, 1, 1, 1D +

s@2, 2, 1, 1D + 2 s@1, 1, 1, 1, 1D + s@2, 1, 1, 1, 1D + s@1, 1, 1, 1, 1, 1D

e@1D2+ 2 e@1D e@2D + e@2D2

+ 2 e@1D e@3D + 2 e@2D e@3D + e@3D2

Sum@e@iD e@jD, 8i, 1, 3<, 8j, 1, 3<DEToS@%, Vars ® 3D

e@1D2+ 2 e@1D e@2D + e@2D2

+ 2 e@1D e@3D + 2 e@2D e@3D + e@3D2

s@2D + s@1, 1D + 2 s@2, 1D + s@2, 2D + 2 s@1, 1, 1D + 3 s@2, 1, 1D + 2 s@2, 2, 1D + s@2, 2, 2D

� EToX

EToX@<f>, <n>D

If f is a symmetric function expressed as a linear combination of e functions, EToX returns the same functionexpressed in the variables x[1], ..., x[n].

EToX@e@1D^2 e@2D, 4D% �� XToE

x@1D3 x@2D + 2 x@1D2 x@2D2+ x@1D x@2D3

+ x@1D3 x@3D + 5 x@1D2 x@2D x@3D +

5 x@1D x@2D2 x@3D + x@2D3 x@3D + 2 x@1D2 x@3D2+ 5 x@1D x@2D x@3D2

+ 2 x@2D2 x@3D2+

x@1D x@3D3+ x@2D x@3D3

+ x@1D3 x@4D + 5 x@1D2 x@2D x@4D + 5 x@1D x@2D2 x@4D + x@2D3 x@4D +

5 x@1D2 x@3D x@4D + 12 x@1D x@2D x@3D x@4D + 5 x@2D2 x@3D x@4D + 5 x@1D x@3D2 x@4D +

5 x@2D x@3D2 x@4D + x@3D3 x@4D + 2 x@1D2 x@4D2+ 5 x@1D x@2D x@4D2

+ 2 x@2D2 x@4D2+

5 x@1D x@3D x@4D2+ 5 x@2D x@3D x@4D2

+ 2 x@3D2 x@4D2+ x@1D x@4D3

+ x@2D x@4D3+ x@3D x@4D3

e@1D2 e@2D

� EToAll

EToAll@<f>, <opts>DOptions@EToAllD=8Targets®8H,P,S,M, X<, Vars®0<

If f is a symmetric function expressed as a linear combination of e functions, EToAll returns the same functionexpressed as linear combinations of the appropriate functions as specified in Targets. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer. Note that the X's will only be displayed if Vars is specified.

symfun31-doc.nb 33

Sum@e@iD e@jD, 8i, 3<, 8j, 3<DEToAll@%D

e@1D2+ 2 e@1D e@2D + e@2D2

+ 2 e@1D e@3D + 2 e@2D e@3D + e@3D2

:e@1D2+ 2 e@1D e@2D + e@2D2

+ 2 e@1D e@3D + 2 e@2D e@3D + e@3D2,

h@1D2+ 2 h@1D3

+ 3 h@1D4+ 2 h@1D5

+ h@1D6- 2 h@1D h@2D - 6 h@1D2 h@2D -

6 h@1D3 h@2D - 4 h@1D4 h@2D + h@2D2+ 4 h@1D h@2D2

+ 4 h@1D2 h@2D2+ 2 h@1D h@3D +

2 h@1D2 h@3D + 2 h@1D3 h@3D - 2 h@2D h@3D - 4 h@1D h@2D h@3D + h@3D2,

p@1D2+ p@1D3

7 p@1D4

36- p@1D p@2D -

2p@1D2 p@2D -

3p@1D3 p@2D -

6p@1D4 p@2D +

2p@1D p@2D2

4p@1D2 p@2D2

3p@1D p@3D +

3p@1D2 p@3D +

9p@1D3 p@3D -

3p@2D p@3D -

3p@1D p@2D p@3D +

s@2D + s@1, 1D + 2 s@2, 1D + s@2, 2D + 2 s@1, 1, 1D + 3 s@2, 1, 1D + 2 s@2, 2, 1D +

s@2, 2, 2D + 3 s@1, 1, 1, 1D + 2 s@2, 1, 1, 1D + s@2, 2, 1, 1D +

2 s@1, 1, 1, 1, 1D + s@2, 1, 1, 1, 1D + s@1, 1, 1, 1, 1, 1D,

m@2D + 2 m@1, 1D + 2 m@2, 1D + m@2, 2D + 6 m@1, 1, 1D + 4 m@2, 1, 1D + 2 m@2, 2, 1D +

m@2, 2, 2D + 14 m@1, 1, 1, 1D + 6 m@2, 1, 1, 1D + 2 m@2, 2, 1, 1D +

20 m@1, 1, 1, 1, 1D + 6 m@2, 1, 1, 1, 1D + 20 m@1, 1, 1, 1, 1, 1D>

Sum@e@iD e@jD, 8i, 3<, 8j, 3<DEToAll@%, Vars ® 2, Targets ® 8P, M, X<D

e@1D2+ 2 e@1D e@2D + e@2D2

+ 2 e@1D e@3D + 2 e@2D e@3D + e@3D2

:e@1D2+ 2 e@1D e@2D + e@2D2

+ 2 e@1D e@3D + 2 e@2D e@3D + e@3D2,

p@1D2+ p@1D3

4- p@1D p@2D -

2p@1D2 p@2D +

4, m@2D + 2 m@1, 1D + 2 m@2, 1D + m@2, 2D,

x@1D2+ 2 x@1D x@2D + 2 x@1D2 x@2D + x@2D2

+ 2 x@1D x@2D2+ x@1D2 x@2D2>

� HTo[EMPSXAll]

� HRestrictionRules

HRestrictionRules@<deg>, <var>D

HRestrictionRules[deg, var] creates a list of rules that express h functions of degree deg in terms of other hfuntions of degree deg when the number of variables is restricted to var. The basis used for this is h[lambda]functions where lambda[[1]]£var (note that this is NOT the same basis generated by HList[deg, var]).

HRestrictionRules@4, 2DHList@4, 2D

9h@4D ® -h@1D4+ h@1D2 h@2D + h@2D2, h@1D h@3D ® -h@1D4

+ 2 h@1D2 h@2D,

h@2D2® h@2D2, h@1D2 h@2D ® h@1D2 h@2D, h@1D4

® h@1D4=

9h@4D, h@1D h@3D, h@2D2=

34 symfun31-doc.nb

HRestrictionRules@6, 1DHList@6, 1D

9h@6D ® h@1D6, h@1D h@5D ® h@1D6, h@2D h@4D ® h@1D6,

h@1D2 h@4D ® h@1D6, h@3D2® h@1D6, h@1D h@2D h@3D ® h@1D6, h@1D3 h@3D ® h@1D6,

h@2D3® h@1D6, h@1D2 h@2D2

® h@1D6, h@1D4 h@2D ® h@1D6, h@1D6® h@1D6=

8h@6D<

� HToE

HToE@<f>, <opts>DOptions@HToED=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of h functions, HToE returns the same functionexpressed as a linear combination of the appropriate e functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

1 + Sum@h@iD, 8i, 1, 6<DHToE@%D% �� EToH

1 + h@1D + h@2D + h@3D + h@4D + h@5D + h@6D

1 + e@1D + e@1D2+ e@1D3

+ e@1D4+ e@1D5

+ e@1D6- e@2D - 2 e@1D e@2D - 3 e@1D2 e@2D -

4 e@1D3 e@2D - 5 e@1D4 e@2D + e@2D2+ 3 e@1D e@2D2

+ 6 e@1D2 e@2D2- e@2D3

+ e@3D +

2 e@1D e@3D + 3 e@1D2 e@3D + 4 e@1D3 e@3D - 2 e@2D e@3D - 6 e@1D e@2D e@3D +

e@3D2- e@4D - 2 e@1D e@4D - 3 e@1D2 e@4D + 2 e@2D e@4D + e@5D + 2 e@1D e@5D - e@6D

1 + h@1D + h@2D + h@3D + h@4D + h@5D + h@6D

1 + Sum@h@iD, 8i, 1, 6<DHToE@%, Vars ® 3D1 + h@1D + h@2D + h@3D + h@4D + h@5D + h@6D

1 + e@1D + e@1D2+ e@1D3

+ e@1D4+ e@1D5

+ e@1D6- e@2D - 2 e@1D e@2D - 3 e@1D2 e@2D -

4 e@1D3 e@2D - 5 e@1D4 e@2D + e@2D2+ 3 e@1D e@2D2

+ 6 e@1D2 e@2D2- e@2D3

+ e@3D +

2 e@1D e@3D + 3 e@1D2 e@3D + 4 e@1D3 e@3D - 2 e@2D e@3D - 6 e@1D e@2D e@3D + e@3D2

� HToM

HToM@<f>, <opts>DOptions@HToMD=8DegreeList®8<,Vars®0<

If f is a symmetric function expressed as a linear combination of h functions, HToM returns the same functionexpressed as a linear combination of the appropriate m functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

symfun31-doc.nb 35

1 + Sum@h@iD, 8i, 1, 6<DHToM@%D% �� MToH

1 + h@1D + h@2D + h@3D + h@4D + h@5D + h@6D

1 + m@1D + m@2D + m@3D + m@4D + m@5D + m@6D + m@1, 1D + m@2, 1D + m@2, 2D + m@3, 1D +

m@3, 2D + m@3, 3D + m@4, 1D + m@4, 2D + m@5, 1D + m@1, 1, 1D + m@2, 1, 1D + m@2, 2, 1D +

m@2, 2, 2D + m@3, 1, 1D + m@3, 2, 1D + m@4, 1, 1D + m@1, 1, 1, 1D + m@2, 1, 1, 1D +

m@2, 2, 1, 1D + m@3, 1, 1, 1D + m@1, 1, 1, 1, 1D + m@2, 1, 1, 1, 1D + m@1, 1, 1, 1, 1, 1D1 + h@1D + h@2D + h@3D + h@4D + h@5D + h@6D

1 + Sum@h@iD, 8i, 1, 6<DHToM@%, Vars ® 3D1 + h@1D + h@2D + h@3D + h@4D + h@5D + h@6D

1 + m@1D + m@2D + m@3D + m@4D + m@5D + m@6D + m@1, 1D + m@2, 1D + m@2, 2D +

m@3, 1D + m@3, 2D + m@3, 3D + m@4, 1D + m@4, 2D + m@5, 1D + m@1, 1, 1D +

m@2, 1, 1D + m@2, 2, 1D + m@2, 2, 2D + m@3, 1, 1D + m@3, 2, 1D + m@4, 1, 1D

� HToP

HToP@<f>, <opts>DOptions@HToPD=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of h functions, HToP returns the same functionexpressed as a linear combination of the appropriate p functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

HToP@h@3D h@2D h@1D - 2 h@2D^2 h@1D^2D% �� PToH

12p@1D6

3p@1D4 p@2D -

4p@1D2 p@2D2

6p@1D3 p@3D +

6p@1D p@2D p@3D

-2 h@1D2 h@2D2+ h@1D h@2D h@3D

HToP@h@3D h@2D h@1D - 2 h@2D^2 h@1D^2, Vars ® 3D

12p@1D6

3p@1D4 p@2D -

4p@1D2 p@2D2

6p@1D3 p@3D +

6p@1D p@2D p@3D

HToP@h@3D h@2D h@1D - 2 h@2D^2 h@1D^2, Vars ® 2D

2p@1D6

2p@1D4 p@2D

� HToS

HToS@<f>, <opts>DOptions@HToSD=8DegreeList®8<, Vars®0<

If f is a symmetric function expressed as a linear combination of h functions, HToS returns the same functionexpressed as a linear combination of the appropriate s functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.

36 symfun31-doc.nb

1 + Sum@h@iD h@jD, 8i, 3<, 8j, 3<DHToS@%D% �� SToH

1 + h@1D2+ 2 h@1D h@2D + h@2D2

+ 2 h@1D h@3D + 2 h@2D h@3D + h@3D2

1 + s@2D + 2 s@3D + 3 s@4D + 2 s@5D + s@6D + s@1, 1D + 2 s@2, 1D +

s@2, 2D + 3 s@3, 1D + 2 s@3, 2D + s@3, 3D + 2 s@4, 1D + s@4, 2D + s@5, 1D

1 + h@1D2+ 2 h@1D h@2D + h@2D2

+ 2 h@1D h@3D + 2 h@2D h@3D + h@3D2

1 + Sum@h@iD h@jD, 8i, 3<, 8j, 3<DHToS@%, Vars ® 1D

1 + h@1D2+ 2 h@1D h@2D + h@2D2

+ 2 h@1D h@3D + 2 h@2D h@3D + h@3D2

1 + s@2D + 2 s@3D + 3 s@4D + 2 s@5D + s@6D

� HToX

HToX@<f>, <n>D

If f is a symmetric function expressed as a linear combination of h functions, HToX returns the same functionexpressed in the variables x[1], ..., x[n].

HToX@h@1D h@2D, 3D

x@1D3+ 2 x@1D2 x@2D + 2 x@1D x@2D2

+ x@2D3+ 2 x@1D2 x@3D +

3 x@1D x@2D x@3D + 2 x@2D2 x@3D + 2 x@1D x@3D2+ 2 x@2D x@3D2

+ x@3D3

� HToAll

HToAll@<f>, <opts>DOptions@HToAllD=8Targets®8E,P,S,M, X<, Vars®0<

If f is a symmetric function expressed as a linear combination of h functions, HToAll returns the same functionexpressed as linear combinations of the appropriate functions as specified in Targets. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer. Note that the X's will only be displayed if Vars is specified.

symfun31-doc.nb 37

1 + Sum@h@iD h@jD, 8i, 3<, 8j, 3<DHToAll@%D

1 + h@1D2+ 2 h@1D h@2D + h@2D2

+ 2 h@1D h@3D + 2 h@2D h@3D + h@3D2

:1 + h@1D2+ 2 h@1D h@2D + h@2D2

+ 2 h@1D h@3D + 2 h@2D h@3D + h@3D2,

1 + e@1D2+ 2 e@1D3

+ 3 e@1D4+ 2 e@1D5

+ e@1D6- 2 e@1D e@2D - 6 e@1D2 e@2D -

6 e@1D3 e@2D - 4 e@1D4 e@2D + e@2D2+ 4 e@1D e@2D2

+ 4 e@1D2 e@2D2+ 2 e@1D e@3D +

2 e@1D2 e@3D + 2 e@1D3 e@3D - 2 e@2D e@3D - 4 e@1D e@2D e@3D + e@3D2,

1 + p@1D2+ p@1D3

7 p@1D4

36+ p@1D p@2D +

2p@1D2 p@2D +

3p@1D3 p@2D +

6p@1D4 p@2D +

2p@1D p@2D2

4p@1D2 p@2D2

3p@1D p@3D +

3p@1D2 p@3D +

9p@1D3 p@3D +

3p@2D p@3D +

3p@1D p@2D p@3D +

1 + s@2D + 2 s@3D + 3 s@4D + 2 s@5D + s@6D + s@1, 1D + 2 s@2, 1D + s@2, 2D +

3 s@3, 1D + 2 s@3, 2D + s@3, 3D + 2 s@4, 1D + s@4, 2D + s@5, 1D,

1 + m@2D + 2 m@3D + 3 m@4D + 2 m@5D + m@6D + 2 m@1, 1D + 4 m@2, 1D + 7 m@2, 2D +

6 m@3, 1D + 6 m@3, 2D + 4 m@3, 3D + 4 m@4, 1D + 3 m@4, 2D + 2 m@5, 1D + 6 m@1, 1, 1D +

10 m@2, 1, 1D + 10 m@2, 2, 1D + 7 m@2, 2, 2D + 8 m@3, 1, 1D + 6 m@3, 2, 1D +

4 m@4, 1, 1D + 14 m@1, 1, 1, 1D + 14 m@2, 1, 1, 1D + 10 m@2, 2, 1, 1D +

8 m@3, 1, 1, 1D + 20 m@1, 1, 1, 1, 1D + 14 m@2, 1, 1, 1, 1D + 20 m@1, 1, 1, 1, 1, 1D>

1 + Sum@h@iD h@jD, 8i, 3<, 8j, 3<DHToAll@%, Vars ® 2, Targets ® 8E, M, X<D

1 + h@1D2+ 2 h@1D h@2D + h@2D2

+ 2 h@1D h@3D + 2 h@2D h@3D + h@3D2

91 + h@1D2+ 2 h@1D h@2D + h@2D2

+ 2 h@1D h@3D + 2 h@2D h@3D + h@3D2,

1 + e@1D2+ 2 e@1D3

+ 3 e@1D4+ 2 e@1D5

+ e@1D6- 2 e@1D e@2D - 6 e@1D2 e@2D -

6 e@1D3 e@2D - 4 e@1D4 e@2D + e@2D2+ 4 e@1D e@2D2

+ 4 e@1D2 e@2D2,

1 + m@2D + 2 m@3D + 3 m@4D + 2 m@5D + m@6D + 2 m@1, 1D + 4 m@2, 1D + 7 m@2, 2D +

6 m@3, 1D + 6 m@3, 2D + 4 m@3, 3D + 4 m@4, 1D + 3 m@4, 2D + 2 m@5, 1D,

1 + x@1D2+ 2 x@1D3

+ 3 x@1D4+ 2 x@1D5

+ x@1D6+ 2 x@1D x@2D + 4 x@1D2 x@2D +

6 x@1D3 x@2D + 4 x@1D4 x@2D + 2 x@1D5 x@2D + x@2D2+ 4 x@1D x@2D2

+ 7 x@1D2 x@2D2+

6 x@1D3 x@2D2+ 3 x@1D4 x@2D2

+ 2 x@2D3+ 6 x@1D x@2D3

+ 6 x@1D2 x@2D3+ 4 x@1D3 x@2D3

3 x@2D4+ 4 x@1D x@2D4

+ 3 x@1D2 x@2D4+ 2 x@2D5

+ 2 x@1D x@2D5+ x@2D6=

Miscellaneous� LR

LRProduct@<Μ>,<Ν>D or s@<Μ>Ds@<Ν>D

Computes the expansion of s[Μ]s[Ν], expressed as a linear combination of s[Λ]'s.

s@<Μ>D<n>

Computes the expression s@ΜDn

s@<Μ>Ds@<Ν>Ds@<Ψ>D... ��Expand

Computes the product s[Μ] s[Ν] s[j]... Products of more than two terms need to be followed by "//Expand" in orderto expand completely. There is no need to type //Expand for s@ < Μ >Dn

38 symfun31-doc.nb

Computes the product s[Μ] s[Ν] s[j]... Products of more than two terms need to be followed by "//Expand" in orderto expand completely. There is no need to type //Expand for s@ < Μ >Dn

LRProduct@82, 1<, 82, 1<D

s@3, 3D + s@4, 2D + s@2, 2, 2D + 2 s@3, 2, 1D + s@4, 1, 1D + s@2, 2, 1, 1D + s@3, 1, 1, 1D

s@2, 1D s@2, 1D

s@3, 3D + s@4, 2D + s@2, 2, 2D + 2 s@3, 2, 1D + s@4, 1, 1D + s@2, 2, 1, 1D + s@3, 1, 1, 1D

s@2, 1D s@2, 1D s@2, 1Ds@2, 1D s@2, 1D s@2, 1D �� Expand

s@2, 1D Hs@3, 3D + s@4, 2D + s@2, 2, 2D + 2 s@3, 2, 1D + s@4, 1, 1D + s@2, 2, 1, 1D + s@3, 1, 1, 1DL

2 s@5, 4D + s@6, 3D + 2 s@3, 3, 3D + 8 s@4, 3, 2D + 4 s@4, 4, 1D +

4 s@5, 2, 2D + 6 s@5, 3, 1D + 2 s@6, 2, 1D + 4 s@3, 2, 2, 2D + 8 s@3, 3, 2, 1D +

9 s@4, 2, 2, 1D + 9 s@4, 3, 1, 1D + 6 s@5, 2, 1, 1D + s@6, 1, 1, 1D +

2 s@2, 2, 2, 2, 1D + 6 s@3, 2, 2, 1, 1D + 4 s@3, 3, 1, 1, 1D + 6 s@4, 2, 1, 1, 1D +

2 s@5, 1, 1, 1, 1D + s@2, 2, 2, 1, 1, 1D + 2 s@3, 2, 1, 1, 1, 1D + s@4, 1, 1, 1, 1, 1Ds@1D s@1D s@1D s@1D s@1D s@1D s@1D s@1D s@1D �� Expand �� Timing

831.45, s@9D + 42 s@5, 4D + 48 s@6, 3D + 27 s@7, 2D + 8 s@8, 1D + 42 s@3, 3, 3D + 168 s@4, 3, 2D +

84 s@4, 4, 1D + 120 s@5, 2, 2D + 162 s@5, 3, 1D + 105 s@6, 2, 1D + 28 s@7, 1, 1D +

84 s@3, 2, 2, 2D + 168 s@3, 3, 2, 1D + 216 s@4, 2, 2, 1D + 216 s@4, 3, 1, 1D +

189 s@5, 2, 1, 1D + 56 s@6, 1, 1, 1D + 42 s@2, 2, 2, 2, 1D + 162 s@3, 2, 2, 1, 1D +

120 s@3, 3, 1, 1, 1D + 189 s@4, 2, 1, 1, 1D + 70 s@5, 1, 1, 1, 1D + 48 s@2, 2, 2, 1, 1, 1D +

105 s@3, 2, 1, 1, 1, 1D + 56 s@4, 1, 1, 1, 1, 1D + 27 s@2, 2, 1, 1, 1, 1, 1D +

28 s@3, 1, 1, 1, 1, 1, 1D + 8 s@2, 1, 1, 1, 1, 1, 1, 1D + s@1, 1, 1, 1, 1, 1, 1, 1, 1D<

s@1D9 �� Timing

831.372, s@9D + 42 s@5, 4D + 48 s@6, 3D + 27 s@7, 2D + 8 s@8, 1D + 42 s@3, 3, 3D + 168 s@4, 3, 2D +

84 s@4, 4, 1D + 120 s@5, 2, 2D + 162 s@5, 3, 1D + 105 s@6, 2, 1D + 28 s@7, 1, 1D +

84 s@3, 2, 2, 2D + 168 s@3, 3, 2, 1D + 216 s@4, 2, 2, 1D + 216 s@4, 3, 1, 1D +

189 s@5, 2, 1, 1D + 56 s@6, 1, 1, 1D + 42 s@2, 2, 2, 2, 1D + 162 s@3, 2, 2, 1, 1D +

120 s@3, 3, 1, 1, 1D + 189 s@4, 2, 1, 1, 1D + 70 s@5, 1, 1, 1, 1D + 48 s@2, 2, 2, 1, 1, 1D +

105 s@3, 2, 1, 1, 1, 1D + 56 s@4, 1, 1, 1, 1, 1D + 27 s@2, 2, 1, 1, 1, 1, 1D +

28 s@3, 1, 1, 1, 1, 1, 1D + 8 s@2, 1, 1, 1, 1, 1, 1, 1D + s@1, 1, 1, 1, 1, 1, 1, 1, 1D<

Finally, a "check" that the calculation is correct:

s@2, 2D s@3, 1D s@1D �� Expand

aaa = SToX@%, 7D;

bbb = HSToX@s@2, 2D, 7DL HSToX@s@3, 1D, 7DL HSToX@s@1D, 7DL �� Expand;

aaa � bbb

s@5, 4D + s@6, 3D + s@3, 3, 3D + 3 s@4, 3, 2D + s@4, 4, 1D + 2 s@5, 2, 2D +

3 s@5, 3, 1D + s@6, 2, 1D + s@3, 2, 2, 2D + 2 s@3, 3, 2, 1D + 3 s@4, 2, 2, 1D +

2 s@4, 3, 1, 1D + 2 s@5, 2, 1, 1D + s@3, 2, 2, 1, 1D + s@4, 2, 1, 1, 1DTrue

symfun31-doc.nb 39

� Bigger examples

LRProduct@84, 3, 2, 1<, 83, 2, 1<D

s@4, 4, 4, 4D + 3 s@5, 4, 4, 3D + 2 s@5, 5, 3, 3D + 3 s@5, 5, 4, 2D + s@5, 5, 5, 1D +

3 s@6, 4, 3, 3D + 3 s@6, 4, 4, 2D + 4 s@6, 5, 3, 2D + 2 s@6, 5, 4, 1D + s@6, 6, 2, 2D +

s@6, 6, 3, 1D + s@7, 3, 3, 3D + 2 s@7, 4, 3, 2D + s@7, 4, 4, 1D + s@7, 5, 2, 2D +

s@7, 5, 3, 1D + s@4, 3, 3, 3, 3D + 3 s@4, 4, 3, 3, 2D + 2 s@4, 4, 4, 2, 2D + 3 s@4, 4, 4, 3, 1D +

3 s@5, 3, 3, 3, 2D + 6 s@5, 4, 3, 2, 2D + 6 s@5, 4, 3, 3, 1D + 6 s@5, 4, 4, 2, 1D +

2 s@5, 5, 2, 2, 2D + 6 s@5, 5, 3, 2, 1D + 3 s@5, 5, 4, 1, 1D + 3 s@6, 3, 3, 2, 2D +

3 s@6, 3, 3, 3, 1D + 3 s@6, 4, 2, 2, 2D + 8 s@6, 4, 3, 2, 1D + 3 s@6, 4, 4, 1, 1D +

4 s@6, 5, 2, 2, 1D + 4 s@6, 5, 3, 1, 1D + s@6, 6, 2, 1, 1D + s@7, 3, 2, 2, 2D + 2 s@7, 3, 3, 2, 1D +

2 s@7, 4, 2, 2, 1D + 2 s@7, 4, 3, 1, 1D + s@7, 5, 2, 1, 1D + s@4, 3, 3, 2, 2, 2D +

2 s@4, 3, 3, 3, 2, 1D + s@4, 4, 2, 2, 2, 2D + 4 s@4, 4, 3, 2, 2, 1D + 3 s@4, 4, 3, 3, 1, 1D +

3 s@4, 4, 4, 2, 1, 1D + s@5, 3, 2, 2, 2, 2D + 4 s@5, 3, 3, 2, 2, 1D + 3 s@5, 3, 3, 3, 1, 1D +

4 s@5, 4, 2, 2, 2, 1D + 8 s@5, 4, 3, 2, 1, 1D + 3 s@5, 4, 4, 1, 1, 1D + 3 s@5, 5, 2, 2, 1, 1D +

3 s@5, 5, 3, 1, 1, 1D + 2 s@6, 3, 2, 2, 2, 1D + 4 s@6, 3, 3, 2, 1, 1D + 4 s@6, 4, 2, 2, 1, 1D +

4 s@6, 4, 3, 1, 1, 1D + 2 s@6, 5, 2, 1, 1, 1D + s@7, 3, 2, 2, 1, 1D + s@7, 3, 3, 1, 1, 1D +

s@7, 4, 2, 1, 1, 1D + s@4, 3, 3, 2, 2, 1, 1D + s@4, 3, 3, 3, 1, 1, 1D + s@4, 4, 2, 2, 2, 1, 1D +

2 s@4, 4, 3, 2, 1, 1, 1D + s@4, 4, 4, 1, 1, 1, 1D + s@5, 3, 2, 2, 2, 1, 1D +

2 s@5, 3, 3, 2, 1, 1, 1D + 2 s@5, 4, 2, 2, 1, 1, 1D + 2 s@5, 4, 3, 1, 1, 1, 1D +

s@5, 5, 2, 1, 1, 1, 1D + s@6, 3, 2, 2, 1, 1, 1D + s@6, 3, 3, 1, 1, 1, 1D + s@6, 4, 2, 1, 1, 1, 1D

� Pieri

Pieri@<Λ>,<k>D

Pieri[Λ, k] computes the list of partitions that are "k-Pieri" over lambda (the partitions that come about by adding kboxes to the Ferrer's diagram of Λ with no more than one box added per column). Corresponds to the special caseof the LR rule where one multiplies by s[n].

Pieri@83, 3, 3<, 6D

889, 3, 3<, 86, 3, 3, 3<, 87, 3, 3, 2<, 88, 3, 3, 1<<

LRProduct@83, 3, 3<, 86<D

s@9, 3, 3D + s@6, 3, 3, 3D + s@7, 3, 3, 2D + s@8, 3, 3, 1D

LRProduct@83, 3, 3<, 81<D

s@4, 3, 3D + s@3, 3, 3, 1D

Pieri@82, 1<, 1D

882, 2<, 83, 1<, 82, 1, 1<<

SMult@82, 1<, 1D

s@2, 2D + s@3, 1D + s@2, 1, 1D

40 symfun31-doc.nb

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