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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).

%I A278289 #71 Sep 25 2020 22:55:51
%S A278289 1,1,16,101376,1190156828672,68978321274090930831360,
%T A278289 40824193474825703180733027309531955200,
%U A278289 440873872874088459550341319780612789503586208384381091840,140992383930585613207663170866505518985873138480180692888967131590224605582721024
%N A278289 Number of standard Young tableaux of skew shape (2n-1,2n-2,...,2,1)/(n-1,n-2,..,2,1).
%D A278289 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Corollary 7.16.3.
%H A278289 Alejandro H. Morales, <a href="/A278289/b278289.txt">Table of n, a(n) for n = 0..22</a>
%H A278289 A. H. Morales, I. Pak and G. Panova, <a href="https://arxiv.org/abs/1610.07561">Asymptotics of the number of standard Young tableaux of skew shape</a>, arXiv:1610.07561 [math.CO], 2016; European Journal of Combinatorics, Vol 70 (2018).
%H A278289 A. H. Morales, I. Pak and G. Panova, <a href="https://arxiv.org/abs/1610.04744">Hook formulas for skew shapes II. Combinatorial proofs and enumerative applications</a>, arXiv:1610.04744 [math.CO], 2016; SIAM Journal of Discrete Mathematics, Vol 31 (2017).
%H A278289 A. H. Morales, I. Pak and M. Tassy, <a href="https://arxiv.org/abs/1805.00992">Asymptotics for the number of standard tableaux of skew shape and for weighted lozenge tilings</a>, arXiv:1805.00992 [math.CO], 2018.
%H A278289 A. H. Morales and D. G. Zhu, <a href="https://arxiv.org/abs/2007.05006">On the Okounkov--Olshanski formula for standard tableaux of skew shapes</a>, arXiv:2007.05006 [math.CO], 2020.
%H A278289 H. Naruse, <a href="http://www.emis.de/journals/SLC/wpapers/s73vortrag/naruse.pdf">Schubert calculus and hook formula</a>, talk slides at 73rd Sém. Lothar. Combin., Strobl, Austria, 2014.
%H A278289 I. Pak, <a href="https://www.math.ucla.edu/~pak/papers/Ribbon1.pdf">Skew shape asymptotics, a case-based introduction</a>, 2020.
%H A278289 Jay Pantone, <a href="http://jaypantone.com/oeis/A278289-439-terms.txt">File with list of n, a(n) for n = 0..438</a> (warning: file size is 100MB)
%F A278289 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).
%F A278289 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]
%e A278289 For n = 3 there are a(2) = 16 standard tableaux of shape (3,2,1)/(1).
%p A278289 a:=proc(k) local lam,mu;
%p A278289 lam:=[seq(2*k-i,i=1..2*k-1)];
%p A278289 mu:=[seq(k-i,i=1..k-1),seq(0,i=1..k)];
%p A278289 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))));
%p A278289 end proc:
%p A278289 seq(a(n),n=0..5);
%Y A278289 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).
%K A278289 nonn
%O A278289 0,3
%A A278289 _Alejandro H. Morales_, Nov 16 2016