Alternating Permutations - MIT Mathematicsrstan/transparencies/altperm-ams.pdf · Simsun...
Transcript of Alternating Permutations - MIT Mathematicsrstan/transparencies/altperm-ams.pdf · Simsun...
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Alternating PermutationsRichard P. Stanley
M.I.T.
Alternating Permutations – p. 1
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Basic definitions
A sequence a1, a2, . . . , ak of distinct integers isalternating if
a1 > a2 < a3 > a4 < · · · ,
and reverse alternating if
a1 < a2 > a3 < a4 > · · · .
Alternating Permutations – p. 2
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Euler numbers
Sn : symmetric group of all permutations of 1, 2, . . . , n
Euler number:
En = #{w ∈ Sn : w is alternating}= #{w ∈ Sn : w is reverse alternating}
E.g., E4 = 5 : 2143, 3142, 3241, 4132, 4231
Alternating Permutations – p. 3
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Euler numbers
Sn : symmetric group of all permutations of 1, 2, . . . , n
Euler number:
En = #{w ∈ Sn : w is alternating}= #{w ∈ Sn : w is reverse alternating}
E.g., E4 = 5 : 2143, 3142, 3241, 4132, 4231
Alternating Permutations – p. 3
![Page 5: Alternating Permutations - MIT Mathematicsrstan/transparencies/altperm-ams.pdf · Simsun permutations (2) b1b2 bk has a double descent if for some 1 < i < n, bi 1 > bi >](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d2f078388c993893a8c4a16/html5/thumbnails/5.jpg)
Euler numbers
Sn : symmetric group of all permutations of 1, 2, . . . , n
Euler number:
En = #{w ∈ Sn : w is alternating}= #{w ∈ Sn : w is reverse alternating}
E.g., E4 = 5 : 2143, 3142, 3241, 4132, 4231
Alternating Permutations – p. 3
![Page 6: Alternating Permutations - MIT Mathematicsrstan/transparencies/altperm-ams.pdf · Simsun permutations (2) b1b2 bk has a double descent if for some 1 < i < n, bi 1 > bi >](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d2f078388c993893a8c4a16/html5/thumbnails/6.jpg)
André’s theorem
Theorem (Désiré André, 1879)
y:=∑
n≥0
Enxn
n!= sec x + tan x
E2n is a secant number.
E2n+1 is a tangent number.
⇒ combinatorial trigonometry
Alternating Permutations – p. 4
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André’s theorem
Theorem (Désiré André, 1879)
y:=∑
n≥0
Enxn
n!= sec x + tan x
E2n is a secant number.
E2n+1 is a tangent number.
⇒ combinatorial trigonometry
Alternating Permutations – p. 4
![Page 8: Alternating Permutations - MIT Mathematicsrstan/transparencies/altperm-ams.pdf · Simsun permutations (2) b1b2 bk has a double descent if for some 1 < i < n, bi 1 > bi >](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d2f078388c993893a8c4a16/html5/thumbnails/8.jpg)
André’s theorem
Theorem (Désiré André, 1879)
y:=∑
n≥0
Enxn
n!= sec x + tan x
E2n is a secant number.
E2n+1 is a tangent number.
⇒ combinatorial trigonometry
Alternating Permutations – p. 4
![Page 9: Alternating Permutations - MIT Mathematicsrstan/transparencies/altperm-ams.pdf · Simsun permutations (2) b1b2 bk has a double descent if for some 1 < i < n, bi 1 > bi >](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d2f078388c993893a8c4a16/html5/thumbnails/9.jpg)
Example of combinatorial trig.
sec2 x = 1 + tan2 x
Equate coefficients of x2n/(2n)!:
n∑
k=0
(
2n
2k
)
E2kE2(n−k)
=n−1∑
k=0
(
2n
2k + 1
)
E2k+1E2(n−k)−1.
Prove combinatorially (exercise).
Alternating Permutations – p. 5
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Example of combinatorial trig.
sec2 x = 1 + tan2 x
Equate coefficients of x2n/(2n)!:
n∑
k=0
(
2n
2k
)
E2kE2(n−k)
=n−1∑
k=0
(
2n
2k + 1
)
E2k+1E2(n−k)−1.
Prove combinatorially (exercise).
Alternating Permutations – p. 5
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Example of combinatorial trig.
sec2 x = 1 + tan2 x
Equate coefficients of x2n/(2n)!:
n∑
k=0
(
2n
2k
)
E2kE2(n−k)
=n−1∑
k=0
(
2n
2k + 1
)
E2k+1E2(n−k)−1.
Prove combinatorially (exercise).
Alternating Permutations – p. 5
![Page 12: Alternating Permutations - MIT Mathematicsrstan/transparencies/altperm-ams.pdf · Simsun permutations (2) b1b2 bk has a double descent if for some 1 < i < n, bi 1 > bi >](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d2f078388c993893a8c4a16/html5/thumbnails/12.jpg)
Proof of André’s theorem
y :=∑
n≥0
Enxn
n!= sec x + tan x
Naive proof.
2En+1 =n∑
k=0
(
n
k
)
EkEn−k, n ≥ 1
⇒ 2y′ = 1 + y2, etc.
(details omitted)
Alternating Permutations – p. 6
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Proof of André’s theorem
y :=∑
n≥0
Enxn
n!= sec x + tan x
Naive proof.
2En+1 =n∑
k=0
(
n
k
)
EkEn−k, n ≥ 1
⇒ 2y′ = 1 + y2, etc.
(details omitted)
Alternating Permutations – p. 6
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Some occurences of Euler numbers
(1) E2n−1 is the number of complete increasingbinary trees on the vertex set[2n + 1] = {1, 2, . . . , 2n + 1}.
Alternating Permutations – p. 7
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Five vertices
1 1
3 34554
2 2
1 1
2 24554
3 3
1
5
1
34435
11
354
534
2 2 2 2
1111
33224 5 5 4 4 5 5 4
2233
1
5
1
3 4 4 35
1 1
3 54
5 34
2222
Slightly more complicated for E2n
Alternating Permutations – p. 8
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Five vertices
1 1
3 34554
2 2
1 1
2 24554
3 3
1
5
1
34435
11
354
534
2 2 2 2
1111
33224 5 5 4 4 5 5 4
2233
1
5
1
3 4 4 35
1 1
3 54
5 34
2222
Slightly more complicated for E2n
Alternating Permutations – p. 8
![Page 17: Alternating Permutations - MIT Mathematicsrstan/transparencies/altperm-ams.pdf · Simsun permutations (2) b1b2 bk has a double descent if for some 1 < i < n, bi 1 > bi >](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d2f078388c993893a8c4a16/html5/thumbnails/17.jpg)
Simsun permutations
(2) b1b2 · · · bk has a double descent if for some1 < i < n,
bi−1 > bi > bi+1.
w = a1a2 · · · an ∈ Sn is a simsun permutation ifthe subsequence with elements 1, 2, . . . , k has nodouble descents, 1 ≤ k ≤ n.
Example. 3241 is not simsun: the subsequencewith 1, 2, 3 is 321.
Theorem (R. Simion & S. Sundaram) Thenumber of simsun permutations in Sn is En+1.
Alternating Permutations – p. 9
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Simsun permutations
(2) b1b2 · · · bk has a double descent if for some1 < i < n,
bi−1 > bi > bi+1.
w = a1a2 · · · an ∈ Sn is a simsun permutation ifthe subsequence with elements 1, 2, . . . , k has nodouble descents, 1 ≤ k ≤ n.
Example. 3241 is not simsun: the subsequencewith 1, 2, 3 is 321.
Theorem (R. Simion & S. Sundaram) Thenumber of simsun permutations in Sn is En+1.
Alternating Permutations – p. 9
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Simsun permutations
(2) b1b2 · · · bk has a double descent if for some1 < i < n,
bi−1 > bi > bi+1.
w = a1a2 · · · an ∈ Sn is a simsun permutation ifthe subsequence with elements 1, 2, . . . , k has nodouble descents, 1 ≤ k ≤ n.
Example. 3241 is not simsun: the subsequencewith 1, 2, 3 is 321.
Theorem (R. Simion & S. Sundaram) Thenumber of simsun permutations in Sn is En+1.
Alternating Permutations – p. 9
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Simsun permutations
(2) b1b2 · · · bk has a double descent if for some1 < i < n,
bi−1 > bi > bi+1.
w = a1a2 · · · an ∈ Sn is a simsun permutation ifthe subsequence with elements 1, 2, . . . , k has nodouble descents, 1 ≤ k ≤ n.
Example. 3241 is not simsun: the subsequencewith 1, 2, 3 is 321.
Theorem (R. Simion & S. Sundaram) Thenumber of simsun permutations in Sn is En+1.
Alternating Permutations – p. 9
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Orbits of mergings
(3) Start with n one-element sets {1}, . . . , {n}.
Merge together two at a time until reaching{1, 2, . . . , n}.
1−2−3−4−5−6, 12−3−4−5−6, 12−34−5−6
125−34−6, 125−346, 123456
Sn acts on these sequences.
Theorem. The number of Sn-orbits is En−1.
Alternating Permutations – p. 10
![Page 22: Alternating Permutations - MIT Mathematicsrstan/transparencies/altperm-ams.pdf · Simsun permutations (2) b1b2 bk has a double descent if for some 1 < i < n, bi 1 > bi >](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d2f078388c993893a8c4a16/html5/thumbnails/22.jpg)
Orbits of mergings
(3) Start with n one-element sets {1}, . . . , {n}.
Merge together two at a time until reaching{1, 2, . . . , n}.
1−2−3−4−5−6, 12−3−4−5−6, 12−34−5−6
125−34−6, 125−346, 123456
Sn acts on these sequences.
Theorem. The number of Sn-orbits is En−1.
Alternating Permutations – p. 10
![Page 23: Alternating Permutations - MIT Mathematicsrstan/transparencies/altperm-ams.pdf · Simsun permutations (2) b1b2 bk has a double descent if for some 1 < i < n, bi 1 > bi >](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d2f078388c993893a8c4a16/html5/thumbnails/23.jpg)
Orbits of mergings
(3) Start with n one-element sets {1}, . . . , {n}.
Merge together two at a time until reaching{1, 2, . . . , n}.
1−2−3−4−5−6, 12−3−4−5−6, 12−34−5−6
125−34−6, 125−346, 123456
Sn acts on these sequences.
Theorem. The number of Sn-orbits is En−1.
Alternating Permutations – p. 10
![Page 24: Alternating Permutations - MIT Mathematicsrstan/transparencies/altperm-ams.pdf · Simsun permutations (2) b1b2 bk has a double descent if for some 1 < i < n, bi 1 > bi >](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d2f078388c993893a8c4a16/html5/thumbnails/24.jpg)
Orbits of mergings
(3) Start with n one-element sets {1}, . . . , {n}.
Merge together two at a time until reaching{1, 2, . . . , n}.
1−2−3−4−5−6, 12−3−4−5−6, 12−34−5−6
125−34−6, 125−346, 123456
Sn acts on these sequences.
Theorem. The number of Sn-orbits is En−1.
Alternating Permutations – p. 10
![Page 25: Alternating Permutations - MIT Mathematicsrstan/transparencies/altperm-ams.pdf · Simsun permutations (2) b1b2 bk has a double descent if for some 1 < i < n, bi 1 > bi >](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d2f078388c993893a8c4a16/html5/thumbnails/25.jpg)
Orbits of mergings
(3) Start with n one-element sets {1}, . . . , {n}.
Merge together two at a time until reaching{1, 2, . . . , n}.
1−2−3−4−5−6, 12−3−4−5−6, 12−34−5−6
125−34−6, 125−346, 123456
Sn acts on these sequences.
Theorem. The number of Sn-orbits is En−1.
Alternating Permutations – p. 10
![Page 26: Alternating Permutations - MIT Mathematicsrstan/transparencies/altperm-ams.pdf · Simsun permutations (2) b1b2 bk has a double descent if for some 1 < i < n, bi 1 > bi >](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d2f078388c993893a8c4a16/html5/thumbnails/26.jpg)
Orbit representatives for n = 5
12−3−4−5 123−4−5 1234−5
12−3−4−5 123−4−5 123−45
12−3−4−5 12−34−5 125−34
12−3−4−5 12−34−5 12−345
12−3−4−5 12−34−5 1234−5
Alternating Permutations – p. 11
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Volume of a polytope
(4) Let En be the convex polytope in Rn defined
by
xi ≥ 0, 1 ≤ i ≤ n
xi + xi+1 ≤ 1, 1 ≤ i ≤ n − 1.
Theorem. The volume of En is En/n!.
Alternating Permutations – p. 12
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Volume of a polytope
(4) Let En be the convex polytope in Rn defined
by
xi ≥ 0, 1 ≤ i ≤ n
xi + xi+1 ≤ 1, 1 ≤ i ≤ n − 1.
Theorem. The volume of En is En/n!.
Alternating Permutations – p. 12
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The “nicest” proof
Triangulate En so that the maximal simplicesσw are indexed by alternating permutationsw ∈ Sn.
Show Vol(σw) = 1/n!.
Alternating Permutations – p. 13
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The “nicest” proof
Triangulate En so that the maximal simplicesσw are indexed by alternating permutationsw ∈ Sn.
Show Vol(σw) = 1/n!.
Alternating Permutations – p. 13
![Page 31: Alternating Permutations - MIT Mathematicsrstan/transparencies/altperm-ams.pdf · Simsun permutations (2) b1b2 bk has a double descent if for some 1 < i < n, bi 1 > bi >](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d2f078388c993893a8c4a16/html5/thumbnails/31.jpg)
Tridiagonal matrices
An n × n matrix M = (mij) is tridiagonal ifmij = 0 whenever |i − j| ≥ 2.
doubly-stochastic: mij ≥ 0, row and columnsums equal 1
Tn: set of n × n tridiagonal doubly stochasticmatrices
Alternating Permutations – p. 14
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Polytope structure of Tn
Easy fact: the map
Tn → Rn−1
M 7→ (m12,m23, . . . ,mn−1,n)
is a (linear) bijection from Tn to En.
Application (Diaconis et al.): random doublystochastic tridiagonal matrices and random walkson Tn
Alternating Permutations – p. 15
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Polytope structure of Tn
Easy fact: the map
Tn → Rn−1
M 7→ (m12,m23, . . . ,mn−1,n)
is a (linear) bijection from Tn to En.
Application (Diaconis et al.): random doublystochastic tridiagonal matrices and random walkson Tn
Alternating Permutations – p. 15
![Page 34: Alternating Permutations - MIT Mathematicsrstan/transparencies/altperm-ams.pdf · Simsun permutations (2) b1b2 bk has a double descent if for some 1 < i < n, bi 1 > bi >](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d2f078388c993893a8c4a16/html5/thumbnails/34.jpg)
Distribution of is(w)
Yesterday: is(w) = length of longest increasingsubsequence of w ∈ Sn
E(n) ∼ 2√
n
For fixed t ∈ R,
limn→∞
Prob
(
isn(w) − 2√
n
n1/6≤ t
)
= F (t),
the Tracy-Widom distribution.
Alternating Permutations – p. 16
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Distribution of is(w)
Yesterday: is(w) = length of longest increasingsubsequence of w ∈ Sn
E(n) ∼ 2√
n
For fixed t ∈ R,
limn→∞
Prob
(
isn(w) − 2√
n
n1/6≤ t
)
= F (t),
the Tracy-Widom distribution.
Alternating Permutations – p. 16
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Analogues of distribution of is(w)
Length of longest alternating subsequence ofw ∈ Sn
Length of longest increasing subsequence ofan alternating permutation w ∈ Sn.
The first is much easier!
Alternating Permutations – p. 17
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Analogues of distribution of is(w)
Length of longest alternating subsequence ofw ∈ Sn
Length of longest increasing subsequence ofan alternating permutation w ∈ Sn.
The first is much easier!
Alternating Permutations – p. 17
![Page 38: Alternating Permutations - MIT Mathematicsrstan/transparencies/altperm-ams.pdf · Simsun permutations (2) b1b2 bk has a double descent if for some 1 < i < n, bi 1 > bi >](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d2f078388c993893a8c4a16/html5/thumbnails/38.jpg)
Analogues of distribution of is(w)
Length of longest alternating subsequence ofw ∈ Sn
Length of longest increasing subsequence ofan alternating permutation w ∈ Sn.
The first is much easier!
Alternating Permutations – p. 17
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Longest alternating subsequences
as(w)= length longest alt. subseq. of w
D(n)=1
n!
∑
w∈Sn
as(w) ∼ ?
w = 56218347 ⇒ as(w) = 5
Alternating Permutations – p. 18
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Definitions of ak(n) and bk(n)
ak(n) = #{w ∈ Sn : as(w) = k}
bk(n) = a1(n) + a2(n) + · · · + ak(n)
= #{w ∈ Sn : as(w) ≤ k}.
Alternating Permutations – p. 19
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The case n = 3
w as(w)
123 1132 2213 3231 2312 3321 2
a1(3) = 1, a2(3) = 3, a3(3) = 2
b1(3) = 1, b2(3) = 4, b3(3) = 6
Alternating Permutations – p. 20
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The case n = 3
w as(w)
123 1132 2213 3231 2312 3321 2
a1(3) = 1, a2(3) = 3, a3(3) = 2
b1(3) = 1, b2(3) = 4, b3(3) = 6
Alternating Permutations – p. 20
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The main lemma
Lemma. ∀w ∈ Sn ∃ alternating subsequence ofmaximal length that contains n.
Corollary.
⇒ ak(n) =n∑
j=1
(
n − 1
j − 1
)
∑
2r+s=k−1
(a2r(j − 1) + a2r+1(j − 1)) as(n − j)
Alternating Permutations – p. 21
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The main lemma
Lemma. ∀w ∈ Sn ∃ alternating subsequence ofmaximal length that contains n.
Corollary.
⇒ ak(n) =n∑
j=1
(
n − 1
j − 1
)
∑
2r+s=k−1
(a2r(j − 1) + a2r+1(j − 1)) as(n − j)
Alternating Permutations – p. 21
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The main generating function
B(x, t)=∑
k,n≥0
bk(n)tkxn
n!
Theorem.
B(x, t) =2/ρ
1 − 1−ρt eρx
− 1
ρ,
where ρ=√
1 − t2.
Alternating Permutations – p. 22
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Formulas for bk(n)
Corollary.
⇒ b1(n) = 1
b2(n) = n
b3(n) = 14(3
n − 2n + 3)
b4(n) = 18(4
n − (2n − 4)2n)
...
no such formulas for longest increasingsubsequences
Alternating Permutations – p. 23
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Formulas for bk(n)
Corollary.
⇒ b1(n) = 1
b2(n) = n
b3(n) = 14(3
n − 2n + 3)
b4(n) = 18(4
n − (2n − 4)2n)
...
no such formulas for longest increasingsubsequences
Alternating Permutations – p. 23
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Mean (expectation) of as(w)
D(n) =1
n!
∑
w∈Sn
as(w) =1
n!
n∑
k=1
k · ak(n),
the expectation of as(w) for w ∈ Sn
Let
A(x, t) =∑
k,n≥0
ak(n)tkxn
n!= (1 − t)B(x, t)
= (1 − t)
(
2/ρ
1 − 1−ρt eρx
− 1
ρ
)
.
Alternating Permutations – p. 24
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Mean (expectation) of as(w)
D(n) =1
n!
∑
w∈Sn
as(w) =1
n!
n∑
k=1
k · ak(n),
the expectation of as(w) for w ∈ Sn
Let
A(x, t) =∑
k,n≥0
ak(n)tkxn
n!= (1 − t)B(x, t)
= (1 − t)
(
2/ρ
1 − 1−ρt eρx
− 1
ρ
)
.
Alternating Permutations – p. 24
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Formula for D(n)
∑
n≥0
D(n)xn =∂
∂tA(x, 1)
=6x − 3x2 + x3
6(1 − x)2
= x +∑
n≥2
4n + 1
6xn.
⇒ D(n) =4n + 1
6, n ≥ 2 (why?)
Compare E(n) ∼ 2√
n.
Alternating Permutations – p. 25
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Formula for D(n)
∑
n≥0
D(n)xn =∂
∂tA(x, 1)
=6x − 3x2 + x3
6(1 − x)2
= x +∑
n≥2
4n + 1
6xn.
⇒ D(n) =4n + 1
6, n ≥ 2 (why?)
Compare E(n) ∼ 2√
n.
Alternating Permutations – p. 25
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Formula for D(n)
∑
n≥0
D(n)xn =∂
∂tA(x, 1)
=6x − 3x2 + x3
6(1 − x)2
= x +∑
n≥2
4n + 1
6xn.
⇒ D(n) =4n + 1
6, n ≥ 2 (why?)
Compare E(n) ∼ 2√
n.Alternating Permutations – p. 25
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Variance of as(w)
V (n)=1
n!
∑
w∈Sn
(
as(w) − 4n + 1
6
)2
, n ≥ 2
the variance of as(n) for w ∈ Sn
Corollary.
V (n) =8
45n − 13
180, n ≥ 4
similar results for higher moments
Alternating Permutations – p. 26
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Variance of as(w)
V (n)=1
n!
∑
w∈Sn
(
as(w) − 4n + 1
6
)2
, n ≥ 2
the variance of as(n) for w ∈ Sn
Corollary.
V (n) =8
45n − 13
180, n ≥ 4
similar results for higher moments
Alternating Permutations – p. 26
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Variance of as(w)
V (n)=1
n!
∑
w∈Sn
(
as(w) − 4n + 1
6
)2
, n ≥ 2
the variance of as(n) for w ∈ Sn
Corollary.
V (n) =8
45n − 13
180, n ≥ 4
similar results for higher moments
Alternating Permutations – p. 26
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A new distribution?
P (t) = limn→∞
Probw∈Sn
(
as(w) − 2n/3√n
≤ t
)
Stanley distribution?
Alternating Permutations – p. 27
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A new distribution?
P (t) = limn→∞
Probw∈Sn
(
as(w) − 2n/3√n
≤ t
)
Stanley distribution?
Alternating Permutations – p. 27
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Limiting distribution
Theorem (Pemantle, Widom, (Wilf)).
limn→∞
Probw∈Sn
(
as(w) − 2n/3√n
≤ t
)
=1√π
∫ t√
45/4
−∞e−s2
ds
(Gaussian distribution)
Alternating Permutations – p. 28
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Limiting distribution
Theorem (Pemantle, Widom, (Wilf)).
limn→∞
Probw∈Sn
(
as(w) − 2n/3√n
≤ t
)
=1√π
∫ t√
45/4
−∞e−s2
ds
(Gaussian distribution)
Alternating Permutations – p. 28
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Umbral enumeration
Umbral formula: involves Ek, where E is anindeterminate (the umbra). Replace Ek with theEuler number Ek. (Technique from 19th century,modernized by Rota et al.)
Example.
(1 + E2)3 = 1 + 3E2 + 3E4 + E6
= 1 + 3E2 + 3E4 + E6
= 1 + 3 · 1 + 3 · 5 + 61
= 80
Alternating Permutations – p. 29
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Umbral enumeration
Umbral formula: involves Ek, where E is anindeterminate (the umbra). Replace Ek with theEuler number Ek. (Technique from 19th century,modernized by Rota et al.)
Example.
(1 + E2)3 = 1 + 3E2 + 3E4 + E6
= 1 + 3E2 + 3E4 + E6
= 1 + 3 · 1 + 3 · 5 + 61
= 80
Alternating Permutations – p. 29
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Another example
(1 + t)E = 1 + Et +
(
E
2
)
t2 +
(
E
3
)
t3 + · · ·
= 1 + Et +1
2E(E − 1)t2 + · · ·
= 1 + E1t +1
2(E2 − E1))t
2 + · · ·
= 1 + t +1
2(1 − 1)t2 + · · ·
= 1 + t + O(t3).
Alternating Permutations – p. 30
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Alt. fixed-point free involutions
fixed-point free involution w ∈ S2n: all cyclesof length two
Let f(n) be the number of alternatingfixed-point free involutions in S2n.
n = 3 : 214365 = (1, 2)(3, 4)(5, 6)
645231 = (1, 6)(2, 4)(3, 5)
f(3) = 2
Alternating Permutations – p. 31
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Alt. fixed-point free involutions
fixed-point free involution w ∈ S2n: all cyclesof length two
Let f(n) be the number of alternatingfixed-point free involutions in S2n.
n = 3 : 214365 = (1, 2)(3, 4)(5, 6)
645231 = (1, 6)(2, 4)(3, 5)
f(3) = 2
Alternating Permutations – p. 31
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Alt. fixed-point free involutions
fixed-point free involution w ∈ S2n: all cyclesof length two
Let f(n) be the number of alternatingfixed-point free involutions in S2n.
n = 3 : 214365 = (1, 2)(3, 4)(5, 6)
645231 = (1, 6)(2, 4)(3, 5)
f(3) = 2
Alternating Permutations – p. 31
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An umbral theorem
Theorem.F (x) :=
∑
n≥0
f(n)xn
=
(
1 + x
1 − x
)(E2+1)/4
Alternating Permutations – p. 32
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An umbral theorem
Theorem.F (x) :=
∑
n≥0
f(n)xn
=
(
1 + x
1 − x
)(E2+1)/4
Alternating Permutations – p. 32
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Proof idea
Proof. Uses representation theory of thesymmetric group Sn.
There is a character χ of Sn (due to H. O.Foulkes) such that for all w ∈ Sn,
χ(w) = 0 or ± Ek.
Now use known results on combinatorialproperties of characters of Sn.
Alternating Permutations – p. 33
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Proof idea
Proof. Uses representation theory of thesymmetric group Sn.
There is a character χ of Sn (due to H. O.Foulkes) such that for all w ∈ Sn,
χ(w) = 0 or ± Ek.
Now use known results on combinatorialproperties of characters of Sn.
Alternating Permutations – p. 33
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Proof idea
Proof. Uses representation theory of thesymmetric group Sn.
There is a character χ of Sn (due to H. O.Foulkes) such that for all w ∈ Sn,
χ(w) = 0 or ± Ek.
Now use known results on combinatorialproperties of characters of Sn.
Alternating Permutations – p. 33
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Ramanujan’s Lost Notebook
Theorem (Ramanujan, Berndt, implicitly) Asx → 0+,
2∑
n≥0
(
1 − x
1 + x
)n(n+1)
∼∑
k≥0
f(k)xk = F (x),
an analytic (non-formal) identity.
Alternating Permutations – p. 34
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A formal identity
Corollary (via Ramanujan, Andrews).
F (x) = 2∑
n≥0
qn
∏nj=1(1 − q2j−1)∏2n+1
j=1 (1 + qj),
where q =(
1−x1+x
)2/3, a formal identity.
Alternating Permutations – p. 35
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Simple result, hard proof
Recall: number of n-cycles in Sn is (n − 1)!.
Theorem. Let b(n) be the number ofalternating n-cycles in Sn. Then if n is odd,
b(n) =1
n
∑
d|nµ(d)(−1)(d−1)/2En/d
∼ En/n.
Alternating Permutations – p. 36
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Simple result, hard proof
Recall: number of n-cycles in Sn is (n − 1)!.
Theorem. Let b(n) be the number ofalternating n-cycles in Sn. Then if n is odd,
b(n) =1
n
∑
d|nµ(d)(−1)(d−1)/2En/d
∼ En/n.
Alternating Permutations – p. 36
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Special case
Corollary. Let p be an odd prime. Then
b(p) =1
p
(
Ep − (−1)(p−1)/2)
.
Combinatorial proof?
Alternating Permutations – p. 37
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Special case
Corollary. Let p be an odd prime. Then
b(p) =1
p
(
Ep − (−1)(p−1)/2)
.
Combinatorial proof?
Alternating Permutations – p. 37
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Inc. subsequences of alt. perms.
Recall: is(w) = length of longest increasingsubsequence of w ∈ Sn. Define
C(n) =1
En
∑
w
is(w),
where w ranges over all En alternatingpermutations in Sn.
Alternating Permutations – p. 38
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β
Little is known, e.g., what is
β = limn→∞
log C(n)
log n?
I.e., C(n) = nβ+o(1).
Compare limn→∞log E(n)
log n = 1/2.
Easy: β ≥ 12.
Alternating Permutations – p. 39
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β
Little is known, e.g., what is
β = limn→∞
log C(n)
log n?
I.e., C(n) = nβ+o(1).
Compare limn→∞log E(n)
log n = 1/2.
Easy: β ≥ 12.
Alternating Permutations – p. 39
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Limiting distribution?
What is the (suitably scaled) limiting distributionof is(w), where w ranges over all alternatingpermutations in Sn?
Is it the Tracy-Widom distribution?
Possible tool: ∃ “umbral analogue” of Gessel’sdeterminantal formula.
Alternating Permutations – p. 40
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Limiting distribution?
What is the (suitably scaled) limiting distributionof is(w), where w ranges over all alternatingpermutations in Sn?
Is it the Tracy-Widom distribution?
Possible tool: ∃ “umbral analogue” of Gessel’sdeterminantal formula.
Alternating Permutations – p. 40
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Limiting distribution?
What is the (suitably scaled) limiting distributionof is(w), where w ranges over all alternatingpermutations in Sn?
Is it the Tracy-Widom distribution?
Possible tool: ∃ “umbral analogue” of Gessel’sdeterminantal formula.
Alternating Permutations – p. 40
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Descent sets
Let w = a1a2 · · · an ∈ Sn. Descent set of w:
D(w) = {i : ai > ai+1} ⊆ {1, . . . , n − 1}
D(4157623) = {1, 4, 5}D(4152736) = {1, 3, 5} (alternating)
D(4736152) = {2, 4, 6} (reverse alternating)
Alternating Permutations – p. 41
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Descent sets
Let w = a1a2 · · · an ∈ Sn. Descent set of w:
D(w) = {i : ai > ai+1} ⊆ {1, . . . , n − 1}
D(4157623) = {1, 4, 5}D(4152736) = {1, 3, 5} (alternating)
D(4736152) = {2, 4, 6} (reverse alternating)
Alternating Permutations – p. 41
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βn(S)
βn(S) = #{w ∈ Sn : D(w) = S}
w D(w)
123 ∅213 {1}312 {1}132 {2}231 {2}321 {1, 2}
β3(∅) = 1, β3(1) = 2
β3(2) = 2, β3(1, 2) = 1
Alternating Permutations – p. 42
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βn(S)
βn(S) = #{w ∈ Sn : D(w) = S}
w D(w)
123 ∅213 {1}312 {1}132 {2}231 {2}321 {1, 2}
β3(∅) = 1, β3(1) = 2
β3(2) = 2, β3(1, 2) = 1
Alternating Permutations – p. 42
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uS
Fix n. Let S ⊆ {1, · · · , n − 1}. Let uS = t1 · · · tn−1,where
ti =
{
a, i 6∈ S
b, i ∈ S.
Example. n = 8, S = {2, 5, 6} ⊆ {1, . . . , 7}uS = abaabba
Alternating Permutations – p. 43
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uS
Fix n. Let S ⊆ {1, · · · , n − 1}. Let uS = t1 · · · tn−1,where
ti =
{
a, i 6∈ S
b, i ∈ S.
Example. n = 8, S = {2, 5, 6} ⊆ {1, . . . , 7}uS = abaabba
Alternating Permutations – p. 43
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A noncommutative gen. function
Ψn(a, b) =∑
S⊆{1,...,n−1}βn(S)uS .
Example. Recall
β3(∅) = 1, β3(1) = 2, β3(2) = 2, β3(1, 2) = 1
ThusΨ3(a, b) = aa + 2ab + 2ba + bb
= (a + b)2 + (ab + ba)
Alternating Permutations – p. 44
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A noncommutative gen. function
Ψn(a, b) =∑
S⊆{1,...,n−1}βn(S)uS .
Example. Recall
β3(∅) = 1, β3(1) = 2, β3(2) = 2, β3(1, 2) = 1
ThusΨ3(a, b) = aa + 2ab + 2ba + bb
= (a + b)2 + (ab + ba)
Alternating Permutations – p. 44
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A noncommutative gen. function
Ψn(a, b) =∑
S⊆{1,...,n−1}βn(S)uS .
Example. Recall
β3(∅) = 1, β3(1) = 2, β3(2) = 2, β3(1, 2) = 1
ThusΨ3(a, b) = aa + 2ab + 2ba + bb
= (a + b)2 + (ab + ba)
Alternating Permutations – p. 44
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The cd-index
Theorem. There exists a noncommutativepolynomial Φn(c, d), called the cd-index of Sn,with nonnegative integer coefficients, such that
Ψn(a, b) = Φn(a + b, ab + ba).
Example. Recall
Ψ3(a, b) = aa+2ab+2ba+ b2 = (a+ b)2 +(ab+ ba).
ThereforeΦ3(c, d) = c2 + d.
Alternating Permutations – p. 45
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The cd-index
Theorem. There exists a noncommutativepolynomial Φn(c, d), called the cd-index of Sn,with nonnegative integer coefficients, such that
Ψn(a, b) = Φn(a + b, ab + ba).
Example. Recall
Ψ3(a, b) = aa+2ab+2ba+ b2 = (a+ b)2 +(ab+ ba).
ThereforeΦ3(c, d) = c2 + d.
Alternating Permutations – p. 45
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Small values of Φn(c, d)
Φ1 = 1
Φ2 = c
Φ3 = c2 + d
Φ4 = c3 + 2cd + 2dc
Φ5 = c4 + 3c2d + 5cdc + 3dc2 + 4d2
Φ6 = c5 + 4c3d + 9c2dc + 9cdc2 + 4dc3
+12cd2 + 10dcd + 12d2c.
Alternating Permutations – p. 46
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Sµ
Let deg c = 1, deg d = 2.
µ: cd-monomial of degree n − 1
Replace each c in µ with 0, each d with 10, andremove final 0. Get the characteristic vector of aset Sµ ⊆ [n − 2].
Example. n = 10:
µ = cd2c2d → 0 · 10 · 10 · 0 · 0 · 10× = 01010001,
the characteristic vector of Sµ = {2, 4, 8} ⊆ [8]
Alternating Permutations – p. 47
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Sµ
Let deg c = 1, deg d = 2.
µ: cd-monomial of degree n − 1
Replace each c in µ with 0, each d with 10, andremove final 0. Get the characteristic vector of aset Sµ ⊆ [n − 2].
Example. n = 10:
µ = cd2c2d → 0 · 10 · 10 · 0 · 0 · 10× = 01010001,
the characteristic vector of Sµ = {2, 4, 8} ⊆ [8]
Alternating Permutations – p. 47
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Sµ
Let deg c = 1, deg d = 2.
µ: cd-monomial of degree n − 1
Replace each c in µ with 0, each d with 10, andremove final 0. Get the characteristic vector of aset Sµ ⊆ [n − 2].
Example. n = 10:
µ = cd2c2d → 0 · 10 · 10 · 0 · 0 · 10× = 01010001,
the characteristic vector of Sµ = {2, 4, 8} ⊆ [8]
Alternating Permutations – p. 47
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Sµ
Let deg c = 1, deg d = 2.
µ: cd-monomial of degree n − 1
Replace each c in µ with 0, each d with 10, andremove final 0. Get the characteristic vector of aset Sµ ⊆ [n − 2].
Example. n = 10:
µ = cd2c2d → 0 · 10 · 10 · 0 · 0 · 10× = 01010001,
the characteristic vector of Sµ = {2, 4, 8} ⊆ [8]
Alternating Permutations – p. 47
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cd-index coefficients
Recall: w = a1a2 · · · an ∈ Sn is a simsunpermutation if the subsequence with elements1, 2, . . . , k has no double descents, 1 ≤ k ≤ n.
Theorem (Simion-Sundaram, variant ofFoata-Schützenberger) The coefficient of µ inΦ(c, d) is equal to the number of simsunpermutations in Sn−1 with descent set Sµ.
Alternating Permutations – p. 48
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cd-index coefficients
Recall: w = a1a2 · · · an ∈ Sn is a simsunpermutation if the subsequence with elements1, 2, . . . , k has no double descents, 1 ≤ k ≤ n.
Theorem (Simion-Sundaram, variant ofFoata-Schützenberger) The coefficient of µ inΦ(c, d) is equal to the number of simsunpermutations in Sn−1 with descent set Sµ.
Alternating Permutations – p. 48
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An example
Example. Φ6 =c5+4c3d+9c2dc+9cdc2+4dc3+12cd2+10dcd+12d2c,
dcd → 10 · 0 · 10× ⇒ Sdcd = {1, 4}
The ten simsun permutations w ∈ S5 withD(w) = {1, 4}:
21354, 21453, 31254, 31452, 41253
41352, 42351, 51243, 51342, 52341,
but not 32451.
Alternating Permutations – p. 49
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An example
Example. Φ6 =c5+4c3d+9c2dc+9cdc2+4dc3+12cd2+10dcd+12d2c,
dcd → 10 · 0 · 10× ⇒ Sdcd = {1, 4}
The ten simsun permutations w ∈ S5 withD(w) = {1, 4}:
21354, 21453, 31254, 31452, 41253
41352, 42351, 51243, 51342, 52341,
but not 32451.
Alternating Permutations – p. 49
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An example
Example. Φ6 =c5+4c3d+9c2dc+9cdc2+4dc3+12cd2+10dcd+12d2c,
dcd → 10 · 0 · 10× ⇒ Sdcd = {1, 4}
The ten simsun permutations w ∈ S5 withD(w) = {1, 4}:
21354, 21453, 31254, 31452, 41253
41352, 42351, 51243, 51342, 52341,
but not 32451.
Alternating Permutations – p. 49
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Two consequences
Theorem. (a) Φn(1, 1) = En (the number ofsimsum permutations w ∈ Sn).
(b) (Niven, de Bruijn) For all S ⊆ {1, . . . , n − 1},
βn(S) ≤ En,
with equality if and only if S = {1, 3, 5, . . . } orS = {2, 4, 6 . . . }
Alternating Permutations – p. 50
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Two consequences
Theorem. (a) Φn(1, 1) = En (the number ofsimsum permutations w ∈ Sn).
(b) (Niven, de Bruijn) For all S ⊆ {1, . . . , n − 1},
βn(S) ≤ En,
with equality if and only if S = {1, 3, 5, . . . } orS = {2, 4, 6 . . . }
Alternating Permutations – p. 50
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An example
Φ5 = 1c4 + 3c2d + 5cdc + 3dc2 + 4d2
1 + 3 + 5 + 3 + 4 = 16 = E5
Alternating Permutations – p. 51
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Alternating Permutations – p. 52