

A336674


Number of positive terms of the OkounkovOlshanski formula for the number of standard tableaux of skew shape (n+3,n+2,...,1)/(n1,n2,...,1).


0



1, 1, 5, 65, 1757, 87129, 7286709, 965911665, 193387756045, 56251615627273, 23021497112124901, 12903943243053179681, 9680994096074346690365, 9530338509606467082850745, 12099590059386455266220499477
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OFFSET

0,3


COMMENTS

a(n) is also the number of semistandard Young tableaux of skew shape (n+3,n+2,...,1)/(n1,n2,...,1) such that the entries in row i are at most i for i=1,...,n+3.
a(n) is also the number of semistandard Young tableaux T of shape (n1,n2,...,1) such that ji < T(i,j) <= n+3 for all cells (i,j).


LINKS



FORMULA

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)!.


EXAMPLE

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)=705=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]].


MAPLE

b := proc(n)
return 2*(1)^n*(14^n)*bernoulli(2*n)/factorial(2*n);
end proc:
a := proc(n)
return factorial(2*n+4)*factorial(2*n+6)*(b(n+1)*b(n+3)b(n+2)^2)/6;
end proc:
seq(a(n), n=0..10);


PROG

(Sage) def b(n):
return 2*(1)^n*(14^n)*bernoulli(2*n)/factorial(2*n) ;
def a(n):
return factorial(2*n+4)*factorial(2*n+6)*(b(n+1)*b(n+3)b(n+2)^2)/6;
[a(i) for i in range(10)]


CROSSREFS

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


KEYWORD

nonn


AUTHOR



STATUS

approved



