OFFSET
0,3
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Corollary 7.16.3.
LINKS
Alejandro H. Morales, Table of n, a(n) for n = 0..22
A. H. Morales, I. Pak and G. Panova, Asymptotics of the number of standard Young tableaux of skew shape, arXiv:1610.07561 [math.CO], 2016; European Journal of Combinatorics, Vol 70 (2018).
A. H. Morales, I. Pak and G. Panova, Hook formulas for skew shapes II. Combinatorial proofs and enumerative applications, arXiv:1610.04744 [math.CO], 2016; SIAM Journal of Discrete Mathematics, Vol 31 (2017).
A. H. Morales, I. Pak and M. Tassy, Asymptotics for the number of standard tableaux of skew shape and for weighted lozenge tilings, arXiv:1805.00992 [math.CO], 2018.
A. H. Morales and D. G. Zhu, On the Okounkov--Olshanski formula for standard tableaux of skew shapes, arXiv:2007.05006 [math.CO], 2020.
H. Naruse, Schubert calculus and hook formula, talk slides at 73rd Sém. Lothar. Combin., Strobl, Austria, 2014.
I. Pak, Skew shape asymptotics, a case-based introduction, 2020.
Jay Pantone, File with list of n, a(n) for n = 0..438 (warning: file size is 100MB)
FORMULA
a(n) = ((3*n^2-n)/2)!*det(1/(lambda[i]-mu[j]-i+j)!), where lambda = (2*n-1,2*n-2,...,1) and mu = (n-1,n-2,...,1,0...,0).
There is a constant c such that log(a(k)) = n*log(n)/2 + c*n + o(n) where n = k*(3*k-1)/2 goes to infinity and -0.2368 <= c <= -0.1648. [updated by Alejandro H. Morales, Aug 29 2020]
EXAMPLE
For n = 3 there are a(2) = 16 standard tableaux of shape (3,2,1)/(1).
MAPLE
a:=proc(k) local lam, mu;
lam:=[seq(2*k-i, i=1..2*k-1)];
mu:=[seq(k-i, i=1..k-1), seq(0, i=1..k)];
factorial(binomial(2*k, 2)-binomial(k, 2))*LinearAlgebra:-Determinant(Matrix(2*k-1, 2*k-1, (i, j)->`if`(lam[i]-mu[j]-i+j<0, 0, 1/factorial(lam[i]-mu[j]-i+j))));
end proc:
seq(a(n), n=0..5);
CROSSREFS
KEYWORD
nonn
AUTHOR
Alejandro H. Morales, Nov 16 2016
STATUS
approved