A296624 - OEIS

%I #20 Dec 20 2017 03:26:00

%S 1,1,4,7,20,37,90,171,378,721,1500,2843,5682,10661,20674

%N 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 The condition lambda >= mu restricts the results to the lower triangular part of the matrix formed by products of all pairs of partitions.

%C 'Multiplicity' signifies that terms like k*s(nu) count as k terms.

%H Wouter Meeussen, <a href="/A296624/a296624.m.txt">Mathematica toolbox for symmetric functions</a>

%e For n=3 we have

%e 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 s(2)*s(1) = s(3) + s(2,1) and

%e s(1,1)*s(1) = s(2,1) + s(1,1,1)

%e for a total of 3+2+2 = 7 terms.

%t Tr/@ Table[Sum[

%t   Length[LRRule[\[Lambda], \[Mu]]], {\[Lambda],

%t    Partitions[n - i]}, {\[Mu],

%t    If[2 i === n, Join[{\[Lambda]}, lesspartitions[\[Lambda]]],

%t     Partitions[i]]}], {n, 14}, {i, 0, Floor[(n)/2]}]; (* Uses functions defined in the 'Toolbox for symmetric functions', see Links. *)

%Y Cf. A296625, A296626.

%K nonn,hard,more

%O 0,3

%A _Wouter Meeussen_, Dec 17 2017