cp's OEIS Frontend

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.

%I A291908 #24 Apr 08 2021 08:50:55
%S A291908 1,16,4362327818240,19265181532031090042534736325278852710400,
%T A291908 830325323503973129435791248069702287019820905338483131168940909920954227594481411031040
%N 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.
%C A291908 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).
%H A291908 E. A. DeWitt, <a href="https://deepblue.lib.umich.edu/handle/2027.42/93841">Identities Relating Schur s-Functions and Q-Functions</a>, Ph.D. thesis, University of Michigan, 2012, 73 pp.
%H A291908 A. H. Morales, I. Pak, G. Panova, <a href="https://arxiv.org/abs/1707.00931">Hook formulas for skew shapes III. Multivariate and product formulas</a>, arXiv:1707.00931 [math.CO], 2017.
%F A291908 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).
%F A291908 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
%e A291908 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).
%p A291908 b:=n->mul(factorial(i),i=1..n-1):
%p A291908 c:=n->mul(doublefactorial(2*i-1),i=1..n-1):
%p A291908 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)):
%p A291908 seq(a(n),n=0..9);
%o A291908 (Sage)
%o A291908 def b(n): return mul([factorial(i) for i in range(1,n)])
%o A291908 def d(n): return factorial(n+1)/(2^((n+1)/2)*factorial((n+1)/2))
%o A291908 def c(n): return mul([d(2*i-1) for i in range(1,n)])
%o A291908 def a(n):
%o A291908     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))
%o A291908 [a(n) for n in range(10)]
%Y A291908 Cf. A000178, A057863, A008793, A291871, A000085, A061343, A005118.
%K A291908 nonn
%O A291908 0,2
%A A291908 _Alejandro H. Morales_, Sep 05 2017