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

1,3

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

Links

Peter J. Taylor, Table of n, a(n) for n = 1..5050

Peter J. Taylor, Python program to compute terms

Wikipedia, Symmetric polynomials

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

Example

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

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}]

Prog

(Python) # See Taylor link

Crossrefs

Row sums are A209670, main diagonal is A209671.

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

Sequence in context: A116547 A013612 A112830 * A189745 A062967 A245211

Adjacent sequences: A209666 A209667 A209668 * A209670 A209671 A209672

Keyword

nonn,tabl

Author

Wouter Meeussen, Mar 11 2012

Status

approved