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
KEYWORD
nonn
AUTHOR
Wouter Meeussen, Feb 09 2020
STATUS
approved