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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 ).

%I A361286 #17 Apr 09 2023 02:31:00
%S A361286 1,2,6,18,50,138,430,1242,3666,10938,34598,108098,338634,1058370
%N 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 ).
%C A361286 All the terms for n >= 1 so far are twice an odd integer.
%C A361286 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.
%C A361286 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.
%e A361286 For n=3,
%e A361286     {3} -> 4 s_{3} + 2 s_{2,1}
%e A361286     {2,1} -> 4 s_{3} + 10 s_{2,1} + 4 s_{1,1,1} and
%e A361286     {1,1,1} -> 2 s_{2,1} + 4 s_{1,1,1}
%e A361286 so a(3) = 4 + 10 + 4 = 18.
%e A361286 Also,
%e A361286 s(3)^2 -> s(6)+s(3;3)+s(4;2)+s(5,1) -> {1,1,1,1} ->{1,1,1,1} ->4
%e A361286   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)
%e A361286          -> {1,1,1,2,1,1,1} -> {1,1,1,4,1,1,1} -> 10
%e A361286 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
%t A361286 (* with 'LRRule' and 'skewschur' defined in http://users.telenet.be/Wouter.Meeussen/ToolBox.nb *)
%t A361286 Tr/@ Table[Coefficient[
%t A361286   Total[skewschur[#, \[Lambda], n] & /@
%t A361286     LRRule[\[Lambda], \[Lambda]]], ss[\[Lambda], n] ], {n,
%t A361286   13}, {\[Lambda], Partitions[n]}];
%t A361286 also Table[Total[
%t A361286   Table[Map[Last, Tally[LRRule[\[Lambda], \[Lambda]]] ]^2, {\[Lambda],
%t A361286      Partitions[n]}], 2], {n, 13}];
%Y A361286 Cf. A067855, A322210.
%K A361286 nonn,more,hard
%O A361286 0,2
%A A361286 _Wouter Meeussen_, Mar 07 2023