A361286 Total over all partitions lambda of n, of factors of s_lambda in the skew Schur function s_( nu/lambda ) with (s_lambda)^2 = Sum( C(nu, lambda, lambda) s_nu ).

1, 2, 6, 18, 50, 138, 430, 1242, 3666, 10938, 34598, 108098, 338634, 1058370

Offset: 0

Author

Wouter Meeussen, Mar 07 2023

Keywords

• nonn
• more
• hard

Comments

All the terms for n >= 1 so far are twice an odd integer.

In terms of Young diagrams, this counts how many original copies one gets by first adding n boxes and then removing n boxes while maintaining an allowed Young diagram shape.

Also a(n) is the total over all partitions n of the multiplicities squared, partition by partition, in the LR-expansion of (s_lambda |- n)^2. Notice that this is different from A067855 where the multipliciteis are first summed over all lambda |-n, and finally squared, then summed.

Examples

			For n=3,
    {3} -> 4 s_{3} + 2 s_{2,1}
    {2,1} -> 4 s_{3} + 10 s_{2,1} + 4 s_{1,1,1} and
    {1,1,1} -> 2 s_{2,1} + 4 s_{1,1,1}
so a(3) = 4 + 10 + 4 = 18.
Also,
s(3)^2 -> s(6)+s(3;3)+s(4;2)+s(5,1) -> {1,1,1,1} ->{1,1,1,1} ->4
  s(2;1)^2 ->s(4;2)+s(4;1;1)+s(3;3)+2 s(3;2;1)+s(3;1;1;1)+s(2;2;2)+s(2;2;1;1)
         -> {1,1,1,2,1,1,1} -> {1,1,1,4,1,1,1} -> 10
s(1;1;1)^2 -> s(2;2;2)+s(2;2;1;1)+s(2;1;1;1;1)+s(1^6) ->{1,1,1,1} ->{1,1,1,1} ->4
		

Crossrefs

Cf. A067855, A322210.

Programs

Mathematica

(* with 'LRRule' and 'skewschur' defined in http://users.telenet.be/Wouter.Meeussen/ToolBox.nb *)
Tr/@ Table[Coefficient[
  Total[skewschur[#, \[Lambda], n] & /@
    LRRule[\[Lambda], \[Lambda]]], ss[\[Lambda], n] ], {n,
  13}, {\[Lambda], Partitions[n]}];
also Table[Total[
  Table[Map[Last, Tally[LRRule[\[Lambda], \[Lambda]]] ]^2, {\[Lambda],
     Partitions[n]}], 2], {n, 13}];