A291871 -id:A291871 - cp's OEIS frontend

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

Alejandro H. Morales, Sep 05 2017

nonn

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.

			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.

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.

Cf. A000178, A008793, A291871.

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 *)

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

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)]

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

A291908 Number of standard Young tableaux of skew shape lambda/mu where lambda is the staircase (4n-1,4n-2,...,2,1) and mu is the square n^n.

Original entry on oeis.org

1, 16, 4362327818240, 19265181532031090042534736325278852710400, 830325323503973129435791248069702287019820905338483131168940909920954227594481411031040

Offset: 0

Alejandro H. Morales, Sep 05 2017

nonn

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).

			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).

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.

Cf. A000178, A057863, A008793, A291871, A000085, A061343, A005118.

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);

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

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A291908 Number of standard Young tableaux of skew shape lambda/mu where lambda is the staircase (4n-1,4n-2,...,2,1) and mu is the square n^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Sage

Formula

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Sage

Formula

A291908 Number of standard Young tableaux of skew shape lambda/mu where lambda is the staircase (4n-1,4n-2,...,2,1) and mu is the square n^n.