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![Page 1: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/1.jpg)
CSM Week 1: Introductory Cross-Disciplinary Seminar
Combinatorial Enumeration
Dave WagnerUniversity of Waterloo
![Page 2: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/2.jpg)
CSM Week 1: Introductory Cross-Disciplinary Seminar
Combinatorial Enumeration
Dave WagnerUniversity of Waterloo
I. The Lagrange Implicit Function Theorem and Exponential Generating Functions
![Page 3: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/3.jpg)
CSM Week 1: Introductory Cross-Disciplinary Seminar
Combinatorial Enumeration
Dave WagnerUniversity of Waterloo
I. The Lagrange Implicit Function Theorem and Exponential Generating Functions
II. A Smorgasbord of Combinatorial Identities
![Page 4: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/4.jpg)
II. A Smorgasbord of Combinatorial Identities
1. Multivariate Lagrange Implicit Function Theorem
![Page 5: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/5.jpg)
II. A Smorgasbord of Combinatorial Identities
1. Multivariate Lagrange Implicit Function Theorem
2. The MacMahon Master Theorem
![Page 6: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/6.jpg)
II. A Smorgasbord of Combinatorial Identities
1. Multivariate Lagrange Implicit Function Theorem
2. The MacMahon Master Theorem
3. Cartier-Foata (Viennot) Heap Inversion
![Page 7: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/7.jpg)
II. A Smorgasbord of Combinatorial Identities
1. Multivariate Lagrange Implicit Function Theorem
2. The MacMahon Master Theorem
3. Cartier-Foata (Viennot) Heap Inversion
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
![Page 8: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/8.jpg)
II. A Smorgasbord of Combinatorial Identities
1. Multivariate Lagrange Implicit Function Theorem
2. The MacMahon Master Theorem
3. Cartier-Foata (Viennot) Heap Inversion
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
5. Kirchhoff’s Matrix Tree Theorem
![Page 9: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/9.jpg)
II. A Smorgasbord of Combinatorial Identities
1. Multivariate Lagrange Implicit Function Theorem
2. The MacMahon Master Theorem
3. Cartier-Foata (Viennot) Heap Inversion
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
5. Kirchhoff’s Matrix Tree Theorem
6. The “Four-Fermion Forest Theorem” (C-J-S-S-S)
![Page 10: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/10.jpg)
![Page 11: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/11.jpg)
1. Multivariate LIFT
Commutative ring K
Indeterminates and
Power series in K[[u]].
),...,,( 21 nuuuu ),...,,( 21 nxxxx
)(),...,(),( 1 uuu nGGF
![Page 12: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/12.jpg)
1. Multivariate LIFT
Commutative ring K
Indeterminates and
Power series in K[[u]].
(a) There are unique power series in K[[x]]
such that for each 1 <= j <= n.
),...,,( 21 nuuuu ),...,,( 21 nxxxx
)(),...,(),( 1 uuu nGGF
)(xjR
),...,( 1 njjj RRGxR
![Page 13: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/13.jpg)
1. Multivariate LIFT
(b) For these power series and for any monomial
(I.J. Good, 1960)
j
i
i
jijn u
G
G
uFRRF
)(
)(det)()(][),...,(][ 1
u
uuGuux ααα
n
nxxx ...21
21αx
)(xjR
![Page 14: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/14.jpg)
![Page 15: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/15.jpg)
2. The MacMahon Master Theorem
Special case of Multivariate LIFT in which each
is a homogeneous linear form.
ninii ucucG ...)( 11u
![Page 16: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/16.jpg)
2. The MacMahon Master Theorem
Special case of Multivariate LIFT in which each
is a homogeneous linear form.
(MacMahon, 1915)
ninii ucucG ...)( 11u
ijiij
n
inini cxxcxc
i
det
1][...][
111
αα xx
![Page 17: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/17.jpg)
2. The MacMahon Master Theorem
This can be rephrased as….
ijij
n
inini cxcxc
i
det
1...][
111
0α
αx
![Page 18: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/18.jpg)
2. The MacMahon Master Theorem
This can be rephrased as….
The matrix represents an endomorphism
on an n-dimensional vector space V.
ijij
n
inini cxcxc
i
det
1...][
111
0α
αx
)( ijcC
![Page 19: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/19.jpg)
2. The MacMahon Master Theorem
This can be rephrased as….
The matrix represents an endomorphism
on an n-dimensional vector space V.
There are induced endomorphisms on the symmetric
powers of V, and on the exterior powers of V.
ijij
n
inini cxcxc
i
det
1...][
111
0α
αx
)( ijcC
CS m
Cm
![Page 20: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/20.jpg)
2. The MacMahon Master Theorem
The traces of these induced endomorphisms satisfy
m
n
inini
mi
xcxcCtrS|| 1
11 ...][α
αx
![Page 21: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/21.jpg)
2. The MacMahon Master Theorem
The traces of these induced endomorphisms satisfy
m
n
inini
mi
xcxcCtrS|| 1
11 ...][α
αx
mmn
m
mijij TCtrTc
0
)1(det
![Page 22: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/22.jpg)
2. The MacMahon Master Theorem
By the MacMahon Master Theorem…
This is called the “Boson-Fermion Correspondence”
1
00
)1(
mm
n
m
m
m
mm TCtrTCtrS
![Page 23: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/23.jpg)
2. The MacMahon Master Theorem
By the MacMahon Master Theorem…
This is called the “Boson-Fermion Correspondence”
(Garoufalidis-Le-Zeilberger, 2006)“quantum” MacMahon Master Theorem.
1
00
)1(
mm
n
m
m
m
mm TCtrTCtrS
![Page 24: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/24.jpg)
![Page 25: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/25.jpg)
3. Cartier-Foata/Viennot Heap Inversion
Another example of the Boson-Fermion Correspondence
arising from symmetric functions….
Countably many indeterminates,...),( 21 xxx
![Page 26: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/26.jpg)
3. Cartier-Foata/Viennot Heap Inversion
Another example of the Boson-Fermion Correspondence
arising from symmetric functions….
Countably many indeterminates
Elementary symmetric functions
k
kiii
iiik xxxe...21
21...)(x
,...),( 21 xxx
![Page 27: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/27.jpg)
3. Cartier-Foata/Viennot Heap Inversion
Another example of the Boson-Fermion Correspondence
arising from symmetric functions….
Countably many indeterminates
Elementary symmetric functions
Complete symmetric functions
k
kiii
iiik xxxe...21
21...)(x
k
kiii
iiik xxxh...21
21...)(x
,...),( 21 xxx
![Page 28: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/28.jpg)
3. Cartier-Foata/Viennot Heap Inversion
Generating functions…
10
1)()(i
ik
kk TxTeTE x
![Page 29: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/29.jpg)
3. Cartier-Foata/Viennot Heap Inversion
Generating functions…
10
1)()(i
ik
kk TxTeTE x
10 1
1)()(
i ik
kk Tx
ThTH x
![Page 30: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/30.jpg)
3. Cartier-Foata/Viennot Heap Inversion
Generating functions…
Clearly
10
1)()(i
ik
kk TxTeTE x
10 1
1)()(
i ik
kk Tx
ThTH x
)(
1)(
TETH
![Page 31: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/31.jpg)
3. Cartier-Foata/Viennot Heap Inversion
Let G=(V,E) be a simple graph.A subset S of V is stable provided that no edge
of G has both ends in S.
![Page 32: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/32.jpg)
3. Cartier-Foata/Viennot Heap Inversion
Let G=(V,E) be a simple graph.A subset S of V is stable provided that no edge
of G has both ends in S.
Introduce indeterminates
The stable set enumerator of G is
}:{ Vvxv x
)(
);(stableS
SGZ xx
![Page 33: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/33.jpg)
3. Cartier-Foata/Viennot Heap Inversion
Let G=(V,E) be a simple graph.A subset S of V is stable provided that no edge of
G has both ends in S.
Introduce indeterminates
The stable set enumerator of G is
(Partition function of a zero-temperature lattice gas on G with repulsive nearest-neighbour interactions.)
}:{ Vvxv x
)(
);(stableS
SGZ xx
![Page 34: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/34.jpg)
3. Cartier-Foata/Viennot Heap Inversion
Let G=(V,E) be a simple graph.
Introduce indeterminates
Say that these commute only for non-adjacent vertices:
if and only if
}:{ Vvxv x
vwwv xxxx Ewv },{
![Page 35: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/35.jpg)
3. Cartier-Foata/Viennot Heap Inversion
Let G=(V,E) be a simple graph.
Introduce indeterminates
Say that these commute only for non-adjacent vertices:
if and only if
Let be the set of all finite strings of vertices, modulo the equivalence relation generated by these commutation relations.
}:{ Vvxv x
vwwv xxxx Ewv },{
/*V
![Page 36: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/36.jpg)
3. Cartier-Foata/Viennot Heap Inversion
(Cartier-Foata, 1969)
This identity is valid for power series with merely partially commutative indeterminates, as above.
/* );(
1
V GZ
xx
![Page 37: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/37.jpg)
3. Cartier-Foata/Viennot Heap Inversion
(Cartier-Foata, 1969)
This identity is valid for power series with merely partially commutative indeterminates, as above.
(There are several variations and generalizations of this.)
(Viennot, 1986)(Krattenthaler, preprint)
/* );(
1
V GZ
xx
![Page 38: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/38.jpg)
![Page 39: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/39.jpg)
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
Let G=(V,E) be a finite connected acyclic directed graph properly drawn in the plane.
![Page 40: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/40.jpg)
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
Let G=(V,E) be a finite connected acyclic directed graph properly drawn in the plane.
Let each edge e be weighted by a value w(e) in some commutative ring K.
![Page 41: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/41.jpg)
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
Let G=(V,E) be a finite connected acyclic directed graph properly drawn in the plane.
Let each edge e be weighted by a value w(e) in some commutative ring K.
For a path P, let w(P) be the product of the weights of the edges of P.
![Page 42: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/42.jpg)
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
Let G=(V,E) be a finite connected acyclic directed graph properly drawn in the plane.
Let each edge e be weighted by a value w(e) in some commutative ring K.
For a path P, let w(P) be the product of the weights of the edges of P.
Fix vertices in that cyclic order around the boundary of the infinite face of G.
1121 ,...,,,...,, ZZZAAA kkk
![Page 43: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/43.jpg)
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
Let G=(V,E) be a finite connected acyclic directed graph properly drawn in the plane.
Let each edge e be weighted by a value w(e) in some commutative ring K.
For a path P, let w(P) be the product of the weights of the edges of P.
Fix vertices in that cyclic order around the boundary of the infinite face of G.
Let be the generating function for
all (directed) paths from A_i to Z_j.
1121 ,...,,,...,, ZZZAAA kkk
ji ZAP
ji PwZAM:
)(),(
![Page 44: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/44.jpg)
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
1A
2A
kA
1Z
1kZ
kZ
![Page 45: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/45.jpg)
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
The generating function for the set of all k-tuples of paths
such that* the paths P_i are internally vertex-disjoint* each P_i goes from A_i to Z_iis
),...,,( 21 kPPP
),(det)()...()(),...,,(
21
21
jiPPP
k ZAMPwPwPwk
![Page 46: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/46.jpg)
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
The generating function for the set of all k-tuples of paths
such that* the paths P_i are internally vertex-disjoint* each P_i goes from A_i to Z_iis
),...,,( 21 kPPP
),(det)()...()(),...,,(
21
21
jiPPP
k ZAMPwPwPwk
![Page 47: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/47.jpg)
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
Application:
vertical edges get weight 1.
horizontal edges (a,b)—(a+1,b) getweight x_b
1
2
3
4
5
6
7
8
x
x
x
x
x
x
x
x
![Page 48: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/48.jpg)
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
1
2
3
4
5
6
7
8
x
x
x
x
x
x
x
x
)1,(a
),( ka
The generating functionfor all paths from to
is a complete symmetricfunction
)1,(a ),( ka
)(xkh
![Page 49: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/49.jpg)
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
1
2
3
4
5
6
7
8
x
x
x
x
x
x
x
x
)1,(a
),( ka
The generating functionfor all paths from to
is a complete symmetricfunction
The path shown is codedby the sequence2 2 4 7 7
)1,(a ),( ka
)(xkh
![Page 50: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/50.jpg)
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
1
2
3
4
5
6
7
8
x
x
x
x
x
x
x
x
Sets of vertex-disjointpaths are encoded bytableaux : 1 1 3 62 2 4 7 73 5 5 8
![Page 51: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/51.jpg)
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
1
2
3
4
5
6
7
8
x
x
x
x
x
x
x
x
Sets of vertex-disjointpaths are encoded bytableaux : 1 1 3 62 2 4 7 73 5 5 8
The generating function fortableaux of a given shapeis a symmetric function…
skew Schur function )(/ xs
![Page 52: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/52.jpg)
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
(dual) Jacobi-Trudy Formula
* When these correspond to the irreducible
representations of the symmetric groups.
)(det)(/ xxji jihs
![Page 53: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/53.jpg)
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
(dual) Jacobi-Trudy Formula
* When these correspond to the irreducible
representations of the symmetric groups.
* They are the minors of “generic” Toeplitz matrices.
)(det)(/ xxji jihs
![Page 54: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/54.jpg)
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5. Kirchhoff’s Matrix Tree Theorem
Let G=(V,E) be a finite connected (multi-)graph.
![Page 56: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/56.jpg)
5. Kirchhoff’s Matrix Tree Theorem
Let G=(V,E) be a finite connected (multi-)graph.
Direct each edge e with ends v and w arbitrarily:
Either v—ew or w—ev.
![Page 57: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/57.jpg)
5. Kirchhoff’s Matrix Tree Theorem
Let G=(V,E) be a finite connected (multi-)graph.
Direct each edge e with ends v and w arbitrarily:
Either v—ew or w—ev.
Define a signed incidence matrix of G to be theV-by-E matrix D with entries
otherwise
ev
ve
Dve
0
1
1
![Page 58: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/58.jpg)
5. Kirchhoff’s Matrix Tree Theorem
Fix indeterminates
}:{ Eeye y
![Page 59: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/59.jpg)
5. Kirchhoff’s Matrix Tree Theorem
Fix indeterminates
Let Y be the E-by-E diagonal matrix
}:{ Eeye y
):( EeydiagY e
![Page 60: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/60.jpg)
5. Kirchhoff’s Matrix Tree Theorem
Fix indeterminates
Let Y be the E-by-E diagonal matrix
The weighted Laplacian matrix of G is
}:{ Eeye y
):( EeydiagY e
*DYDL
![Page 61: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/61.jpg)
5. Kirchhoff’s Matrix Tree Theorem
A graph
![Page 62: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/62.jpg)
5. Kirchhoff’s Matrix Tree Theorem
:= G
-1 0 0 -1 -1 0 0 0 0 -1 -1
1 0 -1 0 0 -1 0 0 0 0 0
0 0 0 0 0 1 -1 0 -1 0 1
0 -1 1 1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 -1 1 0 0
0 1 0 0 0 0 1 1 0 1 0
A signed incidence matrix for it
![Page 63: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/63.jpg)
5. Kirchhoff’s Matrix Tree Theorem
y1 y4 y5 y10 y11 y1
y11 y4
y5 y10
y1 y1 y3 y6
y6 y3 0 0
y11 y6
y6 y7 y9 y11 0 y9 y7
y4 y3 0 y2 y3 y4 0 y2
y5 0 y9 0 y5 y8 y9 y8
y10 0 y7 y2
y8 y2 y7 y8 y10
Its weighted Laplacian matrix
![Page 64: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/64.jpg)
5. Kirchhoff’s Matrix Tree Theorem
Fix indeterminates
Let Y be the E-by-E diagonal matrix
The weighted Laplacian matrix of G is
Fix any “ground vertex”
}:{ Eeye y
):( EeydiagY e
*DYDL
Vv 0
![Page 65: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/65.jpg)
5. Kirchhoff’s Matrix Tree Theorem
Fix indeterminates
Let Y be the E-by-E diagonal matrix
The weighted Laplacian matrix of G is
Fix any “ground vertex”
Let be the submatrix of L obtained by deleting the row and the column indexed by
}:{ Eeye y
):( EeydiagY e
*DYDL
Vv 0
)|( 00 vvL
0v
![Page 66: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/66.jpg)
5. Kirchhoff’s Matrix Tree Theorem
With the notation above…
where the summation is over the set of all spanning trees of G.
T Te
eyvvL )|(det 00
![Page 67: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/67.jpg)
5. Kirchhoff’s Matrix Tree Theorem
With the notation above…
where the summation is over the set of all spanning trees of G.
Proof uses the Binet-Cauchy determinant identity and…
T Te
eyvvL )|(det 00
![Page 68: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/68.jpg)
5. Kirchhoff’s Matrix Tree Theorem
Key Lemma:
Let and with ES VR )(#)(#)(# VRS
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5. Kirchhoff’s Matrix Tree Theorem
Key Lemma:
Let and with
Let M be the square submatrix of D obtained by* deleting rows indexed by vertices in R, and* keeping only columns indexed by edges in S.
ES VR )(#)(#)(# VRS
R
S
M
![Page 70: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/70.jpg)
5. Kirchhoff’s Matrix Tree Theorem
Key Lemma:
Let and with
Let M be the square submatrix of L obtained by* deleting rows indexed by vertices in R, and* keeping only columns indexed by edges in S.
Then if (V,S) is a forest in which each tree has exactly one vertex in R,
and otherwise
ES VR )(#)(#)(# VRS
1det M
0det M
![Page 71: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/71.jpg)
5. Kirchhoff’s Matrix Tree Theorem
With the notation above…
where the summation is over the set of all spanning forests F of G such that each
component of F contains exactly one vertex in R.
“Shorthand” notation:
F
FRRL y)|(det
Fe
eF yy
![Page 72: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/72.jpg)
5. Kirchhoff’s Matrix Tree Theorem
With the notation above…
where the summation is over the set of all spanning forests F of G
F
F FintreeT
TVLI y
:
)(#)det(
![Page 73: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/73.jpg)
5. Kirchhoff’s Matrix Tree Theorem
With the notation above…
where the summation is over the set of all spanning forests F of G
But… we really want a formula without the multiplicities on the RHS….
F
F FintreeT
TVLI y
:
)(#)det(
???F
Fy
![Page 74: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/74.jpg)
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6. The “Four-Fermion Forest Theorem” of C-J-S-S-S
Caracciolo-Jacobsen-Saleur-Sokal-Sportiello (2004)
The generating function for spanning forests of G is
Eijejjiie
F
F yLId ψθψθy )(exp)(
![Page 76: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/76.jpg)
6. The “Four-Fermion Forest Theorem” of C-J-S-S-S
“Shorthand” notation
The greek letters stand for fermionic (anticommuting)
variables. et
cetera
in particular
nnddddddd ...)( 2211ψθ
ijji
02 i
![Page 77: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/77.jpg)
6. The “Four-Fermion Forest Theorem” of C-J-S-S-S
“Shorthand” notation
The greek letters stand for fermionic (anticommuting)
variables.
is an operator – it means keep track only of terms in which each variable occurs exactly once, counting each such term with an appropriate sign.
nnddddddd ...)( 2211ψθ
)(ψθd
![Page 78: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/78.jpg)
6. The “Four-Fermion Forest Theorem” of C-J-S-S-S
For any square matrix M
)exp()()det( ψθψθ MdM
![Page 79: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/79.jpg)
6. The “Four-Fermion Forest Theorem” of C-J-S-S-S
For any square matrix M
“Shorthand” notation
)exp()()det( ψθψθ MdM
j
n
i
n
jijimM
1 1
ψθ
![Page 80: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/80.jpg)
6. The “Four-Fermion Forest Theorem” of C-J-S-S-S
For any square matrix M
Compare with C-J-S-S-S:
)exp()()det( ψθψθ MdM
Eijejjiie
F
F yLId ψθψθy )(exp)(
![Page 81: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/81.jpg)
6. The “Four-Fermion Forest Theorem” of C-J-S-S-S
“Traditionally” each vertex gets a commuting (bosonic) indeterminate vx
![Page 82: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/82.jpg)
6. The “Four-Fermion Forest Theorem” of C-J-S-S-S
“Traditionally” each vertex gets a commuting (bosonic) indeterminate
In C-J-S-S-S this has two anticommuting (fermionic) “superpartners”
vx
v v
![Page 83: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/83.jpg)
6. The “Four-Fermion Forest Theorem” of C-J-S-S-S
“Traditionally” each vertex gets a commuting (bosonic) indeterminate
In C-J-S-S-S this has two anticommuting (fermionic) “superpartners”
and the boson is “integrated out”
vx
v v
![Page 84: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/84.jpg)
6. The “Four-Fermion Forest Theorem” of C-J-S-S-S
“Traditionally” each vertex gets a commuting (bosonic) indeterminate
In C-J-S-S-S this has two anticommuting (fermionic) “superpartners”
and the boson is “integrated out”
v v
![Page 85: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/85.jpg)
6. The “Four-Fermion Forest Theorem” of C-J-S-S-S
“Traditionally” each vertex gets a commuting (bosonic) indeterminate
In C-J-S-S-S this has two anticommuting (fermionic) “superpartners”
and the boson is “integrated out”
The integral is interpreted combinatorially, some very pretty sign-cancellations occur, and only the forests survive, each exactly once.
v v
![Page 86: CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo.](https://reader030.fdocuments.us/reader030/viewer/2022032723/56649d015503460f949d34d1/html5/thumbnails/86.jpg)
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I believe there is a department of mind conducted independent of consciousness, where things are fermented and decocted, so that when they are run off they come clear.
-- James Clerk Maxwell
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