A291879 Number of monomials of the Schubert polynomial of the permutation 351624 tensor 1^n.

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

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.

Sequence in context: A114133 A221233 A173190 * A281450 A354564 A354567

Adjacent sequences: A291876 A291877 A291878 * A291880 A291881 A291882

Keyword

nonn

Author

Alejandro H. Morales, Sep 05 2017

Status

approved