A209669 T(n,k) = count of degree k monomials in the elementary symmetric polynomials e(mu,k) summed over all partitions mu of n.

1, 1, 5, 1, 10, 37, 1, 21, 120, 405, 1, 42, 363, 1644, 5251, 1, 85, 1117, 6814, 27405, 84893, 1, 170, 3360, 27404, 138085, 514248, 1556535, 1, 341, 10164, 111045, 701960, 3145848, 11133493, 33175957, 1, 682, 30520, 445132, 3521405, 18956548, 78337448

Offset: 1

Author

Wouter Meeussen, Mar 11 2012

Keywords

• nonn
• tabl

Comments

T(n,2) = A000975(n+1) because only partitions into parts of 1 and 2 contribute, so that T(n,2) = Sum_{k=0..floor(n/2)} 2^(n-2k). - Peter J. Taylor, Mar 01 2017

Examples

			Table starts as
1;
1,  5;
1, 10,  37;
1, 21, 120,  405;
1, 42, 363, 1644, 5251;
...
For n = 2, k = 2 the partitions of n are 2 and 1+1, which correspond respectively to (xy) contributing 1 and (x+y)*(x+y) contributing 4 for a total of 5. - _Peter J. Taylor_, Mar 01 2017
		

Links

• Peter J. Taylor, Table of n, a(n) for n = 1..5050
• Peter J. Taylor, Python program to compute terms
• Wikipedia, Symmetric polynomials

Crossrefs

Row sums are A209670, main diagonal is A209671.

Second column is A000975 offset by 1. - Peter J. Taylor, Mar 01 2017

Programs

Mathematica

e[n_, v_] := Tr[Times @@@ Select[Subsets[Table[Subscript[x, j], {j, v}]], Length[#] == n &]]; e[par_?PartitionQ, v_] := Times @@ (e[#, v] & /@ par); Table[Tr[(e[#, k] & /@ Partitions[l]) /. Subscript[x, _] -> 1], {l, 10}, {k, l}]

Python

# See Taylor link

Formula

T(n,k) = Sum_{lambda} Product_{i} binomial(k, lambda_i) where the sum is over partitions of n. - Peter J. Taylor, Mar 01 2017