A323655 - OEIS

%I #8 Aug 26 2019 21:07:03

%S 1,1,4,7,19,35,80,149,307,566,1092,1974,3643,6447,11498,19947,34636,

%T 58974,100182,167713,279659,461056,756562,1230104,1990255,3195471,

%U 5105540,8103722,12801925,20107448,31439978,48907179,75755094,116797754,179354540,274253042

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

%C 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 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.

%H Andrew Howroyd, <a href="/A323655/b323655.txt">Table of n, a(n) for n = 0..1000</a>

%F a(2*n) = (A005380(2*n) + A005986(n))/2; a(2*n+1) = A005380(2*n+1)/2. - _Andrew Howroyd_, Aug 26 2019

%e Non-isomorphic representatives of the a(1) = 1 through a(4) = 19 multiset partitions with at most 2 distinct vertices:

%e   {{1}}  {{11}}    {{111}}      {{1111}}

%e          {{12}}    {{122}}      {{1122}}

%e          {{1}{1}}  {{1}{11}}    {{1222}}

%e          {{1}{2}}  {{1}{22}}    {{1}{111}}

%e                    {{2}{12}}    {{11}{11}}

%e                    {{1}{1}{1}}  {{1}{122}}

%e                    {{1}{2}{2}}  {{11}{22}}

%e                                 {{12}{12}}

%e                                 {{1}{222}}

%e                                 {{12}{22}}

%e                                 {{2}{122}}

%e                                 {{1}{1}{11}}

%e                                 {{1}{1}{22}}

%e                                 {{1}{2}{12}}

%e                                 {{1}{2}{22}}

%e                                 {{2}{2}{12}}

%e                                 {{1}{1}{1}{1}}

%e                                 {{1}{1}{2}{2}}

%e                                 {{1}{2}{2}{2}}

%e Non-isomorphic representatives of the a(1) = 1 through a(4) = 19 multiset partitions with at most 2 edges:

%e   {{1}}  {{11}}    {{111}}    {{1111}}

%e          {{12}}    {{122}}    {{1122}}

%e          {{1}{1}}  {{123}}    {{1222}}

%e          {{1}{2}}  {{1}{11}}  {{1233}}

%e                    {{1}{22}}  {{1234}}

%e                    {{1}{23}}  {{1}{111}}

%e                    {{2}{12}}  {{11}{11}}

%e                               {{1}{122}}

%e                               {{11}{22}}

%e                               {{12}{12}}

%e                               {{1}{222}}

%e                               {{12}{22}}

%e                               {{1}{233}}

%e                               {{12}{33}}

%e                               {{1}{234}}

%e                               {{12}{34}}

%e                               {{13}{23}}

%e                               {{2}{122}}

%e                               {{3}{123}}

%e Inequivalent representatives of the a(4) = 19 matrices:

%e   [4] [2 2] [1 3]

%e .

%e   [1] [1 0] [1 0] [0 1] [2] [2 0] [1 1] [1 1]

%e   [3] [1 2] [0 3] [1 2] [2] [0 2] [1 1] [0 2]

%e .

%e   [1] [1 0] [1 0] [1 0] [0 1]

%e   [1] [1 0] [0 1] [0 1] [0 1]

%e   [2] [0 2] [1 1] [0 2] [1 1]

%e .

%e   [1] [1 0] [1 0]

%e   [1] [1 0] [0 1]

%e   [1] [0 1] [0 1]

%e   [1] [0 1] [0 1]

%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o 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

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Jan 22 2019

%E Terms a(11) and beyond from _Andrew Howroyd_, Aug 26 2019