A361286 - OEIS

%I #17 Apr 09 2023 02:31:00

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

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

%C 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 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 For n=3,

%e     {3} -> 4 s_{3} + 2 s_{2,1}

%e     {2,1} -> 4 s_{3} + 10 s_{2,1} + 4 s_{1,1,1} and

%e     {1,1,1} -> 2 s_{2,1} + 4 s_{1,1,1}

%e so a(3) = 4 + 10 + 4 = 18.

%e Also,

%e s(3)^2 -> s(6)+s(3;3)+s(4;2)+s(5,1) -> {1,1,1,1} ->{1,1,1,1} ->4

%e   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          -> {1,1,1,2,1,1,1} -> {1,1,1,4,1,1,1} -> 10

%e 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 (* with 'LRRule' and 'skewschur' defined in http://users.telenet.be/Wouter.Meeussen/ToolBox.nb *)

%t Tr/@ Table[Coefficient[

%t   Total[skewschur[#, \[Lambda], n] & /@

%t     LRRule[\[Lambda], \[Lambda]]], ss[\[Lambda], n] ], {n,

%t   13}, {\[Lambda], Partitions[n]}];

%t also Table[Total[

%t   Table[Map[Last, Tally[LRRule[\[Lambda], \[Lambda]]] ]^2, {\[Lambda],

%t      Partitions[n]}], 2], {n, 13}];

%Y Cf. A067855, A322210.

%K nonn,more,hard

%O 0,2

%A _Wouter Meeussen_, Mar 07 2023