A291871 -id:A291871

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A291879

Number of monomials of the Schubert polynomial of the permutation 351624 tensor 1^n.

+0
1

1, 8, 6720, 561120560, 4557185891241984, 3571558033324129373292768, 269111599998006391761541640176800000, 1945556482213500279178010210766074095827609600000, 1347912754604769492992184400055703948513202427323999206349209600

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The permutation 351624 tensor 1^n is the permutation whose permutation matrix is obtained from that of 351624 by replacing each 1 with an n X n identity matrix.

LINKS

Table of n, a(n) for n=0..8.

A. H. Morales, I. Pak, G. Panova, Hook formulas for skew shapes III. Multivariate and product formulas, arXiv:1707.00931 [math.CO], 2017.

R. P. Stanley, Some Schubert shenanigans, arXiv:1704.00851 [math.CO], 2017.

FORMULA

a(n) = b(n)^5*b(3*n)^2*b(5*n)/(b(2*n)^4*b(4*n)^2) where b(n) = 1!*2!*...*(n-1)! is a superfactorial A000178(n-1). [corrected by Vaclav Kotesovec, Apr 08 2021]

a(n) = c(n)*b(3*n)^2*b(6*n)/((7*n^2)!*b(2*n)^2*b(4*n)^2) where b(n) = 1!*2!*...*(n-1)! is a superfactorial A000178(n-1) and c(n) = A291871.

a(n) ~ exp(1/6) * 3^(9*n^2 - 1/6) * 5^(25*n^2/2 - 1/12) / (A^2 * n^(1/6) * 2^(40*n^2 - 2/3)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2021

EXAMPLE

For n=1 we have that a(1)=8 since the Schubert polynomial of 351624 equals the following sum of eight monomials: x0^3*x1^3*x2 + x0^3*x1^2*x2^2 + x0^2*x1^3*x2^2 + x0^3*x1^3*x3 + x0^3*x1^2*x2*x3 + x0^2*x1^3*x2*x3 + x0^3*x1^2*x3^2 + x0^2*x1^3*x3^2.

MATHEMATICA

Table[BarnesG[n + 1]^5 * BarnesG[3*n + 1]^2 * BarnesG[5*n + 1] / (BarnesG[2*n + 1]^4 * BarnesG[4*n + 1]^2), {n, 0, 10}] (* Vaclav Kotesovec, Apr 08 2021 *)

PROG

(Sage) def b(n): return mul([factorial(i) for i in range(1, n)])

def a(n): return b(n)^5*b(3*n)^2*b(5*n)/(b(2*n)^4*b(4*n)^2)

[a(n) for n in range(10)]

(PARI) b(n) = prod(k=1, n-1, k!);

a(n) = b(n)^5*b(3*n)^2*b(5*n)/(b(2*n)^4*b(4*n)^2); \\ Michel Marcus, Sep 07 2017

CROSSREFS

Cf. A000178, A008793, A291871.

KEYWORD

nonn

AUTHOR

Alejandro H. Morales, Sep 05 2017

STATUS

approved

A291908

Number of standard Young tableaux of skew shape lambda/mu where lambda is the staircase (4*n-1,4*n-2,...,2,1) and mu is the square n^n.

+0
1

1, 16, 4362327818240, 19265181532031090042534736325278852710400, 830325323503973129435791248069702287019820905338483131168940909920954227594481411031040

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The number of standard Young tableaux of a fixed skew shape has a determinantal formula, the Jacobi-Trudi formula. It is rare when a family of skew shapes has a product formula for the number of standard Young tableaux. This product formula has independently been proved using P-Schur functions (by DeWitt) and using the Naruse hook-length formula for skew shapes (by Morales, Pak and Panova).

LINKS

Table of n, a(n) for n=0..4.

E. A. DeWitt, Identities Relating Schur s-Functions and Q-Functions, Ph.D. thesis, University of Michigan, 2012, 73 pp.

A. H. Morales, I. Pak, G. Panova, Hook formulas for skew shapes III. Multivariate and product formulas, arXiv:1707.00931 [math.CO], 2017.

FORMULA

a(n) = (binomial(4*n,2)-n^2)!*b(n)^3*b(3*n)*c(n)*c(3*n)/(b(2*n)^3*c(2*n)^2*c(4*n)) where b(n) = 1!*2!*...*(n-1)! is the superfactorial A000178(n-1), and c(n) = 1!!*3!!*...*(2*n-3)!! is super doublefactorial A057863(n-1).

a(n) ~ sqrt(Pi) * 3^(9*n^2 - 3*n/2 - 1/24) * 7^(7*n^2 - 2*n + 1/2) * exp(7*n^2/2 - 2*n + 23/56) * n^(7*n^2 - 2*n + 7/8) / (A^(3/2) * 2^(33*n^2 - 6*n - 7/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2021

EXAMPLE

a(1)=16 since there are 16 standard Young tableaux of skew shape 321/1 since this is the same as the number of standard Young tableaux of straight shape 321 given by the hook-length formula: 16 = 6!/(3^2*5).

MAPLE

b:=n->mul(factorial(i), i=1..n-1):

c:=n->mul(doublefactorial(2*i-1), i=1..n-1):

a:=n->factorial(binomial(4*n, 2)-n^2)*b(n)^3*b(3*n)*c(n)*c(3*n)/(b(2*n)^3*c(2*n)^2*c(4*n)):

seq(a(n), n=0..9);

PROG

(Sage)

def b(n): return mul([factorial(i) for i in range(1, n)])

def d(n): return factorial(n+1)/(2^((n+1)/2)*factorial((n+1)/2))

def c(n): return mul([d(2*i-1) for i in range(1, n)])

def a(n):

return factorial(binomial(4*n, 2)-n^2)*b(n)^3*b(3*n)*c(n)*c(3*n)/(b(2*n)^3*c(2*n)^2*c(4*n))