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A296624 a(n) is the total multiplicity of all products of Schur functions s(lambda)*s(mu) with partition lambda >= mu and size(lambda) + size(mu)= n.

%I A296624 #20 Dec 20 2017 03:26:00
%S A296624 1,1,4,7,20,37,90,171,378,721,1500,2843,5682,10661,20674
%N A296624 a(n) is the total multiplicity of all products of Schur functions s(lambda)*s(mu) with partition lambda >= mu and size(lambda) + size(mu)= n.
%C A296624 The condition lambda >= mu restricts the results to the lower triangular part of the matrix formed by products of all pairs of partitions.
%C A296624 'Multiplicity' signifies that terms like k*s(nu) count as k terms.
%H A296624 Wouter Meeussen, <a href="/A296624/a296624.m.txt">Mathematica toolbox for symmetric functions</a>
%e A296624 For n=3 we have
%e A296624 s(3)*s(0) = s(3); s(2,1)*s(0) = s(2,1); s(1,1,1)*s(0) = s(1,1,1)
%e A296624 s(2)*s(1) = s(3) + s(2,1) and
%e A296624 s(1,1)*s(1) = s(2,1) + s(1,1,1)
%e A296624 for a total of 3+2+2 = 7 terms.
%t A296624 Tr/@ Table[Sum[
%t A296624   Length[LRRule[\[Lambda], \[Mu]]], {\[Lambda],
%t A296624    Partitions[n - i]}, {\[Mu],
%t A296624    If[2 i === n, Join[{\[Lambda]}, lesspartitions[\[Lambda]]],
%t A296624     Partitions[i]]}], {n, 14}, {i, 0, Floor[(n)/2]}]; (* Uses functions defined in the 'Toolbox for symmetric functions', see Links. *)
%Y A296624 Cf. A296625, A296626.
%K A296624 nonn,hard,more
%O A296624 0,3
%A A296624 _Wouter Meeussen_, Dec 17 2017