cp's OEIS Frontend

A332302 Squared length of sum of e_lambda e_lambda', where e_lambda is an elementary symmetric function and lambda ranges over all partitions of n and lambda' is the adjoint (or transpose) of lambda.

%I A332302 #33 Mar 24 2022 16:32:43
%S A332302 1,4,5,9,13,21,29,50,66,98,134,191,255,355,468,633,829,1117,1438,1895,
%T A332302 2432,3156,4021,5163,6520,8292,10406,13108,16345,20438,25320,31491,
%U A332302 38797,47890,58737,72105,87991,107473,130577,158686,192021,232328,279993,337391,405112,486212,581806,695763
%N A332302 Squared length of sum of e_lambda e_lambda', where e_lambda is an elementary symmetric function and lambda ranges over all partitions of n and lambda' is the adjoint (or transpose) of lambda.
%C A332302 Similar to A067855, but with the elementary symmetric function instead of the Schur function. Note that A067855 describes (s_lambda)^2 which equals the count for (s_lambda . s_lambda'). This is not the case for the other symmetric functions. Squared length of sum of (e_lambda)^2 is simply A000041 (the partition numbers).
%C A332302 The result is identical for the homogenous and the power sum symmetric functions h_lambda and p_lambda since all three can be written as products: e_lambda = Product_{i=1..n} e(lambda_i).
%e A332302 For n = 4, we get a(4) = 9 since
%e A332302 e(4)e(1,1,1,1) = e(4,1,1,1,1);
%e A332302 e(3,1)e(2,1,1) = e(3,2,1,1);
%e A332302 e(2,2)e(2,2)   = e(2,2,2,2);
%e A332302 e(2,1,1)e(3,1) = e(3,2,1,1);
%e A332302 e(1,1,1,1)e(4) = e(4,1,1,1,1);
%e A332302 summing to 2 e(4,1,1,1,1) + 2 e(3,2,1,1) + e(2,2,2,2)
%e A332302 with coefficient vector (2,2,1) and length squared 2^2 + 2^2 + 1^2 = 9.
%t A332302 Table[aa = Reverse[Sort[Join[#, TransposePartition[#]]]]&/@ Partitions[n]; (#.#)&@ Map[Last, Tally[aa]], {n, 48}]
%Y A332302 Cf. A000041, A067855.
%K A332302 nonn
%O A332302 1,2
%A A332302 _Wouter Meeussen_, Feb 09 2020