A209673 a(n) = count of monomials, of degree k=n, in the Schur symmetric polynomials s(mu,k) summed over all partitions mu of n.

1, 1, 4, 19, 116, 751, 5552, 43219, 366088, 3245311, 30569012, 299662672, 3079276708, 32773002718, 362512238272, 4136737592323, 48773665308176, 591313968267151, 7375591544495636, 94340754464144215, 1237506718985945656, 16608519982801477908, 228013066931927465872

Offset: 0

Author

Wouter Meeussen, Mar 11 2012

Keywords

• nonn

Comments

Main diagonal of triangle A191714.

a(n) is also the number of semistandard Young tableaux of size and maximal entry n. - Christian Stump, Oct 09 2015

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• Wikipedia, Symmetric Polynomials
• FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux

Crossrefs

Cf. A191714, A209664, A209665, A209666, A209667, A209668, A209669, A209670, A209671, A209672, A209673.

Main diagonal of A210391. - Alois P. Heinz, Mar 22 2012

Programs

Mathematica

(* see A191714 *)
Tr /@ Table[(stanley[#, l] & /@ Partitions[l]), {l, 11}]
(* or *)
Table[SeriesCoefficient[1/((1-x)^(n*(n+1)/2) * (1+x)^(n*(n-1)/2)), {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Aug 06 2025 *)

Extensions

a(12)-a(22) from Alois P. Heinz, Mar 11 2012

Typo in Mathematica program fixed by Vaclav Kotesovec, Mar 19 2015