A323656 Number of non-isomorphic multiset partitions of weight n with exactly 2 distinct vertices, or with exactly 2 (not necessarily distinct) edges.

0, 0, 2, 4, 14, 28, 69, 134, 285, 536, 1050, 1918, 3566, 6346, 11363, 19771, 34405, 58677, 99797, 167223, 279032, 460264, 755560, 1228849, 1988680, 3193513, 5103104, 8100712, 12798207, 20102883, 31434374, 48900337, 75746745, 116787611, 179342230, 274238159

Offset: 0

Author

Gus Wiseman, Jan 22 2019

Keywords

• nonn

Comments

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Also the number of nonnegative integer matrices with only two columns, no zero rows or columns, and sum of entries equal to n, up to row and column permutations.

Examples

			Non-isomorphic representatives of the a(2) = 2 through a(4) = 14 multiset partitions with exactly 2 distinct vertices:
  {{12}}    {{122}}      {{1122}}
  {{1}{2}}  {{1}{22}}    {{1222}}
            {{2}{12}}    {{1}{122}}
            {{1}{2}{2}}  {{11}{22}}
                         {{12}{12}}
                         {{1}{222}}
                         {{12}{22}}
                         {{2}{122}}
                         {{1}{1}{22}}
                         {{1}{2}{12}}
                         {{1}{2}{22}}
                         {{2}{2}{12}}
                         {{1}{1}{2}{2}}
                         {{1}{2}{2}{2}}
Non-isomorphic representatives of the a(2) = 2 through a(4) = 14 multiset partitions with exactly 2 edges:
  {{1}{1}}  {{1}{11}}  {{1}{111}}
  {{1}{2}}  {{1}{22}}  {{11}{11}}
            {{1}{23}}  {{1}{122}}
            {{2}{12}}  {{11}{22}}
                       {{12}{12}}
                       {{1}{222}}
                       {{12}{22}}
                       {{1}{233}}
                       {{12}{33}}
                       {{1}{234}}
                       {{12}{34}}
                       {{13}{23}}
                       {{2}{122}}
                       {{3}{123}}
Inequivalent representatives of the a(4) = 14 matrices:
  [2 2] [1 3]
.
  [1 0] [1 0] [0 1] [2 0] [1 1] [1 1]
  [1 2] [0 3] [1 2] [0 2] [1 1] [0 2]
.
  [1 0] [1 0] [1 0] [0 1]
  [1 0] [0 1] [0 1] [0 1]
  [0 2] [1 1] [0 2] [1 1]
.
  [1 0] [1 0]
  [1 0] [0 1]
  [0 1] [0 1]
  [0 1] [0 1]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000041, A007716, A052847, A054974, A120733, A316980, A323654, A323655, A323656.

Programs

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={concat(0, (EulerT(vector(n, k, k+1)) + EulerT(vector(n, k, if(k%2, 0, (k+6)\4))))/2 - EulerT(vector(n,k,1)))} \\ Andrew Howroyd, Aug 26 2019

Formula

a(n) = A323655(n) - A000041(n). - Andrew Howroyd, Aug 26 2019

Extensions

Terms a(11) and beyond from Andrew Howroyd, Aug 26 2019