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

0,5

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.

Links

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

Formula

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

Example

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]

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

Crossrefs

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

Sequence in context: A260890 A022663 A304967 * A092481 A099508 A141577

Adjacent sequences: A323651 A323652 A323653 * A323655 A323656 A323657

Keyword

nonn

Author

Gus Wiseman, Jan 22 2019

Extensions

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

Status

approved