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 A336674 Number of positive terms of the Okounkov-Olshanski formula for the number of standard tableaux of skew shape (n+3,n+2,...,1)/(n-1,n-2,...,1). 0

%I #18 Jul 31 2020 18:06:11

%S 1,1,5,65,1757,87129,7286709,965911665,193387756045,56251615627273,

%T 23021497112124901,12903943243053179681,9680994096074346690365,

%U 9530338509606467082850745,12099590059386455266220499477

%N Number of positive terms of the Okounkov-Olshanski formula for the number of standard tableaux of skew shape (n+3,n+2,...,1)/(n-1,n-2,...,1).

%C a(n) is also the number of semistandard Young tableaux of skew shape (n+3,n+2,...,1)/(n-1,n-2,...,1) such that the entries in row i are at most i for i=1,...,n+3.

%C a(n) is also the number of semistandard Young tableaux T of shape (n-1,n-2,...,1) such that j-i < T(i,j) <= n+3 for all cells (i,j).

%H 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 shape</a>, arXiv:2007.05006 [math.CO], 2020.

%F a(n) = ((2*n+4)!*(2*n+6)!/3!)*(b(n+1)*b(n+3)-b(n+2)^2) where b(n)=A110501(n)/(2*n)!.

%e For n=2 the a(2)=5 semistandard Young tableaux of skew shape (5,4,3,2,1)/(1) are determined by their first column which are [1,2,3,4], [1,2,3,5], [1,2,4,5], [1,3,4,5], and [2,3,4,5]. Also, the a(2)=5 semistandard Young tableaux of shape (1) with entries between 0 and 5 are [1], [2], [3], [4], and [5]. Also, the a(3)=70-5=65 are the semistandard Young tableaux of shape (2,1) with entries at most 6 excluding the five tableaux whose entry in the first row and first column is 1: [[1,1],[2]], [[1,1],[3]], [[1,1],[4]], and [[1,1],[5]].

%p b := proc(n)

%p return 2*(-1)^n*(1-4^n)*bernoulli(2*n)/factorial(2*n);

%p end proc:

%p a := proc(n)

%p return factorial(2*n+4)*factorial(2*n+6)*(b(n+1)*b(n+3)-b(n+2)^2)/6;

%p end proc:

%p seq(a(n),n=0..10);

%o (Sage) def b(n):

%o return 2*(-1)^n*(1-4^n)*bernoulli(2*n)/factorial(2*n) ;

%o def a(n):

%o return factorial(2*n+4)*factorial(2*n+6)*(b(n+1)*b(n+3)-b(n+2)^2)/6;

%o [a(i) for i in range(10)]

%Y A110501, A005700 gives the number of terms of the Naruse hook length formula for the same skew shape.

%K nonn

%O 0,3

%A _Alejandro H. Morales_, Jul 29 2020

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Last modified August 14 03:23 EDT 2024. Contains 375146 sequences. (Running on oeis4.)