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A214043 Count of Laurent monomials (including multiplicities), in the Symplectic Schur symmetric polynomials s(mu, n) summed over all partitions mu of n. 0

%I #48 Jan 26 2018 03:54:15

%S 2,15,134,1589,20162,293580,4519916,75850054,1334978228,24987138510,

%T 487322528552,9968005618302,211338028257280,4658444968474433,

%U 105985325960653194,2492041019432287042,60271996071301852442,1500054086883728030496

%N Count of Laurent monomials (including multiplicities), in the Symplectic Schur symmetric polynomials s(mu, n) summed over all partitions mu of n.

%H T. Amdeberhan, <a href="http://arxiv.org/abs/1207.4045">Theorems, problems and conjectures</a>, arXiv:1207.4045 [math.RT], 2012-2015.

%e For n = 2, partition = (1, 1), the Symplectic Schur is: x_1*x_2 + x_1/x_2 + x_2/x_1 + 1/(x_1*x_2) + 1. There are five terms here. Partition (2) contributes another ten terms, including the term 1 twice. So a(2) = 5+10 = 15. [Extended by _Andrey Zabolotskiy_, Jan 24 2018]

%t s[mu_,n_] := Expand[Simplify[Det[Table[x[j]^(mu[[i]]+n-i+1) - x[j]^(-mu[[j]]-n+i-1), {i,n}, {j,n}]] / Det[Table[x[j]^(n-i+1) - x[j]^(-n+i-1), {i,n}, {j,n}]]]];

%t Table[Sum[s[PadRight[mu,n], n] /. {x[_]->1}, {mu, IntegerPartitions[n]}], {n, 5}]

%t (* _Andrey Zabolotskiy_, Jan 24 2018 *)

%K nonn

%O 1,1

%A _T. Amdeberhan_, Jul 13 2012

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