cp's OEIS Frontend

A323656 Number of non-isomorphic multiset partitions of weight n with exactly 2 distinct vertices, or with exactly 2 (not necessarily distinct) edges.

%I A323656 #8 Aug 26 2019 21:44:48
%S A323656 0,0,2,4,14,28,69,134,285,536,1050,1918,3566,6346,11363,19771,34405,
%T A323656 58677,99797,167223,279032,460264,755560,1228849,1988680,3193513,
%U A323656 5103104,8100712,12798207,20102883,31434374,48900337,75746745,116787611,179342230,274238159
%N A323656 Number of non-isomorphic multiset partitions of weight n with exactly 2 distinct vertices, or with exactly 2 (not necessarily distinct) edges.
%C A323656 The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
%C A323656 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.
%H A323656 Andrew Howroyd, <a href="/A323656/b323656.txt">Table of n, a(n) for n = 0..1000</a>
%F A323656 a(n) = A323655(n) - A000041(n). - _Andrew Howroyd_, Aug 26 2019
%e A323656 Non-isomorphic representatives of the a(2) = 2 through a(4) = 14 multiset partitions with exactly 2 distinct vertices:
%e A323656   {{12}}    {{122}}      {{1122}}
%e A323656   {{1}{2}}  {{1}{22}}    {{1222}}
%e A323656             {{2}{12}}    {{1}{122}}
%e A323656             {{1}{2}{2}}  {{11}{22}}
%e A323656                          {{12}{12}}
%e A323656                          {{1}{222}}
%e A323656                          {{12}{22}}
%e A323656                          {{2}{122}}
%e A323656                          {{1}{1}{22}}
%e A323656                          {{1}{2}{12}}
%e A323656                          {{1}{2}{22}}
%e A323656                          {{2}{2}{12}}
%e A323656                          {{1}{1}{2}{2}}
%e A323656                          {{1}{2}{2}{2}}
%e A323656 Non-isomorphic representatives of the a(2) = 2 through a(4) = 14 multiset partitions with exactly 2 edges:
%e A323656   {{1}{1}}  {{1}{11}}  {{1}{111}}
%e A323656   {{1}{2}}  {{1}{22}}  {{11}{11}}
%e A323656             {{1}{23}}  {{1}{122}}
%e A323656             {{2}{12}}  {{11}{22}}
%e A323656                        {{12}{12}}
%e A323656                        {{1}{222}}
%e A323656                        {{12}{22}}
%e A323656                        {{1}{233}}
%e A323656                        {{12}{33}}
%e A323656                        {{1}{234}}
%e A323656                        {{12}{34}}
%e A323656                        {{13}{23}}
%e A323656                        {{2}{122}}
%e A323656                        {{3}{123}}
%e A323656 Inequivalent representatives of the a(4) = 14 matrices:
%e A323656   [2 2] [1 3]
%e A323656 .
%e A323656   [1 0] [1 0] [0 1] [2 0] [1 1] [1 1]
%e A323656   [1 2] [0 3] [1 2] [0 2] [1 1] [0 2]
%e A323656 .
%e A323656   [1 0] [1 0] [1 0] [0 1]
%e A323656   [1 0] [0 1] [0 1] [0 1]
%e A323656   [0 2] [1 1] [0 2] [1 1]
%e A323656 .
%e A323656   [1 0] [1 0]
%e A323656   [1 0] [0 1]
%e A323656   [0 1] [0 1]
%e A323656   [0 1] [0 1]
%o A323656 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A323656 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
%Y A323656 Cf. A000041, A007716, A052847, A054974, A120733, A316980, A323654, A323655, A323656.
%K A323656 nonn
%O A323656 0,3
%A A323656 _Gus Wiseman_, Jan 22 2019
%E A323656 Terms a(11) and beyond from _Andrew Howroyd_, Aug 26 2019