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A323656
Number of non-isomorphic multiset partitions of weight n with exactly 2 distinct vertices, or with exactly 2 (not necessarily distinct) edges.
5
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
0,3
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
FORMULA
a(n) = A323655(n) - A000041(n). - Andrew Howroyd, Aug 26 2019
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 22 2019
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Aug 26 2019
STATUS
approved