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A278289 Number of standard Young tableaux of skew shape (2n-1,2n-2,...,2,1)/(n-1,n-2,..,2,1). 1
1, 1, 16, 101376, 1190156828672, 68978321274090930831360, 40824193474825703180733027309531955200, 440873872874088459550341319780612789503586208384381091840, 140992383930585613207663170866505518985873138480180692888967131590224605582721024 (list; graph; refs; listen; history; text; internal format)
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.
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
Cf. A005118; for even n the number of terms in Naruse hook length formula is given by A181119 (Corollary 8.1 in arXiv:1610.04744).
Sequence in context: A368326 A308507 A144830 * A298202 A364777 A332090
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified July 24 03:05 EDT 2024. Contains 374575 sequences. (Running on oeis4.)