A061343 -id:A061343 - cp's OEIS frontend

A291871 Number of standard Young tableaux of skew shape (3n^(2n), 2*n^n)/(n^n).

1, 42, 14617044842400, 5458228515594914179387748450655273588000, 864891828322912925373153355728411014930091519471102108791040960580578545212124160000

Offset: 0

Alejandro H. Morales, Sep 04 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 a combinatorial model for the Selberg integral (by Kim and Oh) and using the Naruse hook-length formula for skew shapes (by Morales, Pak and Panova).

			a(1)=42 since there are 42 standard Young tableaux of skew shape 332/1 since this is the same as the number of standard Young tableaux of straight shape 332 given by the hook-length formula: 42 = 8!/(2^2*3*4^2*5).

J. S. Kim and S. Oh, The Selberg integral and Young books, arXiv:1409.1317 [math.CO], 2014.
J. S. Kim and S. Oh, The Selberg integral and Young books, J. Combin. Theory Ser. A 145 (2017), 1-24.
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, A008793, A291879, A291908, A000085, A061343.

b:=n->mul(factorial(i),i=1..n-1):
a:=n->factorial(7*n^2)*b(n)^5*b(5*n)/(b(2*n)^2*b(6*n)):
seq(a(i),i=0..9);

b[n_] := Product[i!, {i, n - 1}]; Table[(7 n^2)!*b[n]^5*b[5 n]/(b[2 n]^2*b[6 n]), {n, 0, 4}] (* Michael De Vlieger, Sep 10 2017 *)

def b(n): return mul([factorial(i) for i in range(1,n)])
def a(n): return factorial(7*n^2)*b(n)^5*b(5*n)/(b(2*n)^2*b(6*n))
[a(n) for n in range(10)]

a(n) = (7*n^2)!*b(n)^5*b(5*n)/(b(2*n)^2*b(6*n)) where b(n) = 1!*2!*...*(n-1)! is a superfactorial A000178(n-1).
a(n) = (7*n^2)!*c(n)*b(n)^2*b(2*n)*b(5*n)/(b(6*n)*b(3*n)) where b(n) = 1!*2!*...*(n-1)! is a superfactorial A000178(n-1) and c(n) = A008793.
log a(n) = 7*n^2*log(n) + (75/2 - 15*log(2) - 18*log(3) + 25/2*log(5) + 7*log(7))*n^2 + O(n*log(n)). (See Example 6.2 in Morales et al.)
a(n) ~ sqrt(Pi) * 5^(25*n^2/2 - 1/12) * 7^(7*n^2 + 1/2) * exp(7*n^2/2 + 1/4) * n^(7*n^2 + 3/4) / (A^3 * 2^(22*n^2 - 3/4) * 3^(18*n^2 - 1/12)), 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

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A291871 Number of standard Young tableaux of skew shape (3n^(2n), 2*n^n)/(n^n).

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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.

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cp's OEIS Frontend

A291871 Number of standard Young tableaux of skew shape (3*n^(2*n), 2*n^n)/(n^n).

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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.

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A291871 Number of standard Young tableaux of skew shape (3n^(2n), 2*n^n)/(n^n).

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.