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.

1, 4, 5, 9, 13, 21, 29, 50, 66, 98, 134, 191, 255, 355, 468, 633, 829, 1117, 1438, 1895, 2432, 3156, 4021, 5163, 6520, 8292, 10406, 13108, 16345, 20438, 25320, 31491, 38797, 47890, 58737, 72105, 87991, 107473, 130577, 158686, 192021, 232328, 279993, 337391, 405112, 486212, 581806, 695763

Offset: 1

Author

Wouter Meeussen, Feb 09 2020

Keywords

• nonn

Comments

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).

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).

Examples

			For n = 4, we get a(4) = 9 since
e(4)e(1,1,1,1) = e(4,1,1,1,1);
e(3,1)e(2,1,1) = e(3,2,1,1);
e(2,2)e(2,2)   = e(2,2,2,2);
e(2,1,1)e(3,1) = e(3,2,1,1);
e(1,1,1,1)e(4) = e(4,1,1,1,1);
summing to 2 e(4,1,1,1,1) + 2 e(3,2,1,1) + e(2,2,2,2)
with coefficient vector (2,2,1) and length squared 2^2 + 2^2 + 1^2 = 9.
		

Crossrefs

Cf. A000041, A067855.

Programs

Mathematica

Table[aa = Reverse[Sort[Join[#, TransposePartition[#]]]]&/@ Partitions[n]; (#.#)&@ Map[Last, Tally[aa]], {n, 48}]