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 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
EXAMPLE
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]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 22 2019
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Aug 26 2019
STATUS
approved