A296626 -id:A296626 - cp's OEIS frontend

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.

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

Offset: 0

Wouter Meeussen, Dec 17 2017

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

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

			For n=3 we have
s(3)*s(0) = s(3); s(2,1)*s(0) = s(2,1); s(1,1,1)*s(0) = s(1,1,1)
s(2)*s(1) = s(3) + s(2,1) and
s(1,1)*s(1) = s(2,1) + s(1,1,1)
for a total of 3+2+2 = 7 terms.

Wouter Meeussen, Mathematica toolbox for symmetric functions

Cf. A296625, A296626.

Tr/@ Table[Sum[
  Length[LRRule[\[Lambda], \[Mu]]], {\[Lambda],
   Partitions[n - i]}, {\[Mu],
   If[2 i === n, Join[{\[Lambda]}, lesspartitions[\[Lambda]]],
    Partitions[i]]}], {n, 14}, {i, 0, Floor[(n)/2]}]; (* Uses functions defined in the 'Toolbox for symmetric functions', see Links. *)

A296625 a(n) is the total multiplicity of all products of Schur functions s(lambda)*s(lambda) with lambda a partition of n.

Original entry on oeis.org

1, 2, 6, 16, 42, 106, 268, 660, 1618, 3922, 9438, 22540, 53528, 126358

Offset: 0

Wouter Meeussen, Dec 17 2017

Diagonal of the matrix formed by products of all pairs of partitions.
Conjecture: a(n) is the number of domino tilings of diagrams of integer partitions of 2n. - Gus Wiseman, Feb 25 2018
The above conjecture is not true, see A304662. - Alois P. Heinz, May 22 2018

			for n=2,
s(2)*s(2) = s(4) + s(3,1) + s(2,2) and
s(1,1) * s(1,1) = s(2,2) + s(2,1,1) + s(1,1,1,1)
for 6 terms in total.

Cf. A296624, A296626, A304662.

Table[Sum[Length[LRRule[\[Lambda], \[Lambda]]], {\[Lambda], Partitions[n]}], {n, 0, 7}];
(* Uses the Mathematica toolbox for Symmetric Functions from A296624. *)

a(n) = A304662(n) for n < 7. - Alois P. Heinz, May 22 2018

a(13)-a(14) from Wouter Meeussen, Nov 22 2018

cp's OEIS Frontend

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.

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