A323654 Number of non-isomorphic multiset partitions of weight n with no constant parts and only two distinct vertices.

1, 0, 1, 1, 3, 3, 8, 9, 20, 26, 50, 69, 125, 177, 301, 440, 717, 1055, 1675, 2471, 3835, 5660, 8627, 12697, 19095, 27978, 41581, 60650, 89244, 129490, 188925, 272676, 394809, 566882, 815191, 1164510, 1664295, 2365698, 3361844, 4756030, 6723280, 9468138, 13319299

Offset: 0

Author

Gus Wiseman, Jan 22 2019

Keywords

• nonn

Comments

First differs from A304967 at a(10) = 50, A304967(10) = 49.

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 positive integer matrices with only two columns and sum of entries equal to n, up to row and column permutations.

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(7) = 9 multiset partitions:
  {{12}}  {{122}}  {{1122}}    {{11222}}    {{111222}}      {{1112222}}
                   {{1222}}    {{12222}}    {{112222}}      {{1122222}}
                   {{12}{12}}  {{12}{122}}  {{122222}}      {{1222222}}
                                            {{112}{122}}    {{112}{1222}}
                                            {{12}{1122}}    {{12}{11222}}
                                            {{12}{1222}}    {{12}{12222}}
                                            {{122}{122}}    {{122}{1122}}
                                            {{12}{12}{12}}  {{122}{1222}}
                                                            {{12}{12}{122}}
Inequivalent representatives of the a(8) = 20 matrices:
  [4 4] [3 5] [2 6] [1 7]
.
  [1 1] [1 1] [1 1] [2 1] [2 1] [1 2] [1 2] [3 1] [2 2] [2 2] [1 3]
  [3 3] [2 4] [1 5] [2 3] [1 4] [2 3] [1 4] [1 3] [2 2] [1 3] [1 3]
.
  [1 1] [1 1] [1 1] [1 1]
  [1 1] [1 1] [2 1] [1 2]
  [2 2] [1 3] [1 2] [1 2]
.
  [1 1]
  [1 1]
  [1 1]
  [1 1]
		

Links

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

Crossrefs

Cf. A003293, A007716, A052847, A054974, A120733, A321407, A321760, A323655, A323656.

Programs

PARI

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

Formula

a(2*n) = (A052847(2*n) + A003293(n))/2; a(2*n+1) = A052847(2*n+1)/2. - Andrew Howroyd, Aug 26 2019

Extensions

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