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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.
0
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
1,2
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).
EXAMPLE
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.
MATHEMATICA
Table[aa = Reverse[Sort[Join[#, TransposePartition[#]]]]&/@ Partitions[n]; (#.#)&@ Map[Last, Tally[aa]], {n, 48}]
CROSSREFS
Sequence in context: A282467 A318980 A226622 * A287209 A038099 A243166
KEYWORD
nonn
AUTHOR
Wouter Meeussen, Feb 09 2020
STATUS
approved