A323655 Number of non-isomorphic multiset partitions of weight n with at most 2 distinct vertices, or with at most 2 (not necessarily distinct) edges.

1, 1, 4, 7, 19, 35, 80, 149, 307, 566, 1092, 1974, 3643, 6447, 11498, 19947, 34636, 58974, 100182, 167713, 279659, 461056, 756562, 1230104, 1990255, 3195471, 5105540, 8103722, 12801925, 20107448, 31439978, 48907179, 75755094, 116797754, 179354540, 274253042

Offset

0,3

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 one or two columns, no zero rows or columns, and sum of entries equal to n, up to row and column permutations.

Links

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

Formula

a(2*n) = (A005380(2*n) + A005986(n))/2; a(2*n+1) = A005380(2*n+1)/2. - Andrew Howroyd, Aug 26 2019

Example

Non-isomorphic representatives of the a(1) = 1 through a(4) = 19 multiset partitions with at most 2 distinct vertices:
  {{1}}  {{11}}    {{111}}      {{1111}}
         {{12}}    {{122}}      {{1122}}
         {{1}{1}}  {{1}{11}}    {{1222}}
         {{1}{2}}  {{1}{22}}    {{1}{111}}
                   {{2}{12}}    {{11}{11}}
                   {{1}{1}{1}}  {{1}{122}}
                   {{1}{2}{2}}  {{11}{22}}
                                {{12}{12}}
                                {{1}{222}}
                                {{12}{22}}
                                {{2}{122}}
                                {{1}{1}{11}}
                                {{1}{1}{22}}
                                {{1}{2}{12}}
                                {{1}{2}{22}}
                                {{2}{2}{12}}
                                {{1}{1}{1}{1}}
                                {{1}{1}{2}{2}}
                                {{1}{2}{2}{2}}
Non-isomorphic representatives of the a(1) = 1 through a(4) = 19 multiset partitions with at most 2 edges:
  {{1}}  {{11}}    {{111}}    {{1111}}
         {{12}}    {{122}}    {{1122}}
         {{1}{1}}  {{123}}    {{1222}}
         {{1}{2}}  {{1}{11}}  {{1233}}
                   {{1}{22}}  {{1234}}
                   {{1}{23}}  {{1}{111}}
                   {{2}{12}}  {{11}{11}}
                              {{1}{122}}
                              {{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) = 19 matrices:
  [4] [2 2] [1 3]
.
  [1] [1 0] [1 0] [0 1] [2] [2 0] [1 1] [1 1]
  [3] [1 2] [0 3] [1 2] [2] [0 2] [1 1] [0 2]
.
  [1] [1 0] [1 0] [1 0] [0 1]
  [1] [1 0] [0 1] [0 1] [0 1]
  [2] [0 2] [1 1] [0 2] [1 1]
.
  [1] [1 0] [1 0]
  [1] [1 0] [0 1]
  [1] [0 1] [0 1]
  [1] [0 1] [0 1]

Prog

(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+6)\4))))/2)} \\ Andrew Howroyd, Aug 26 2019

Crossrefs

Cf. A007716, A052847, A005380, A054974, A005986, A120733, A316980, A323654, A323655, A323656.

Sequence in context: A102991 A298350 A291596 * A062306 A323105 A006130

Adjacent sequences: A323652 A323653 A323654 * A323656 A323657 A323658

Keyword

nonn

Author

Gus Wiseman, Jan 22 2019

Extensions

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

Status

approved