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

0,3

Comments

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.

Links

Table of n, a(n) for n=0..14.

Wouter Meeussen, Mathematica toolbox for symmetric functions

Example

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.

Mathematica

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. *)

Crossrefs

Cf. A296625, A296626.

Sequence in context: A147065 A049947 A266822 * A296663 A066345 A355357

Adjacent sequences: A296621 A296622 A296623 * A296625 A296626 A296627

Keyword

nonn,hard,more

Author

Wouter Meeussen, Dec 17 2017

Status

approved