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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.

%I A209673 #48 Aug 06 2025 07:21:42
%S A209673 1,1,4,19,116,751,5552,43219,366088,3245311,30569012,299662672,
%T A209673 3079276708,32773002718,362512238272,4136737592323,48773665308176,
%U A209673 591313968267151,7375591544495636,94340754464144215,1237506718985945656,16608519982801477908,228013066931927465872
%N 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.
%C A209673 Main diagonal of triangle A191714.
%C A209673 a(n) is also the number of semistandard Young tableaux of size and maximal entry n. - _Christian Stump_, Oct 09 2015
%H A209673 Alois P. Heinz, <a href="/A209673/b209673.txt">Table of n, a(n) for n = 0..500</a>
%H A209673 Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomials">Symmetric Polynomials</a>
%H A209673 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/SemistandardTableaux">Semistandard Young tableaux</a>
%t A209673 (* see A191714 *)
%t A209673 Tr /@ Table[(stanley[#, l] & /@ Partitions[l]), {l, 11}]
%t A209673 (* or *)
%t A209673 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 *)
%Y A209673 Cf. A191714, A209664, A209665, A209666, A209667, A209668, A209669, A209670, A209671, A209672, A209673.
%Y A209673 Main diagonal of A210391. - _Alois P. Heinz_, Mar 22 2012
%K A209673 nonn
%O A209673 0,3
%A A209673 _Wouter Meeussen_, Mar 11 2012
%E A209673 a(12)-a(22) from _Alois P. Heinz_, Mar 11 2012
%E A209673 Typo in Mathematica program fixed by _Vaclav Kotesovec_, Mar 19 2015