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

2, 15, 134, 1589, 20162, 293580, 4519916, 75850054, 1334978228, 24987138510, 487322528552, 9968005618302, 211338028257280, 4658444968474433, 105985325960653194, 2492041019432287042, 60271996071301852442, 1500054086883728030496

Offset

1,1

Links

Table of n, a(n) for n=1..18.

T. Amdeberhan, Theorems, problems and conjectures, arXiv:1207.4045 [math.RT], 2012-2015.

Example

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]

Mathematica

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}]]]];
Table[Sum[s[PadRight[mu, n], n] /. {x[_]->1}, {mu, IntegerPartitions[n]}], {n, 5}]
(* Andrey Zabolotskiy, Jan 24 2018 *)

Crossrefs

Sequence in context: A143924 A217483 A371579 * A347993 A215922 A285449

Adjacent sequences: A214040 A214041 A214042 * A214044 A214045 A214046

Keyword

nonn

Author

T. Amdeberhan, Jul 13 2012

Status

approved