cp's OEIS Frontend

A322784 Number of multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.

%I A322784 #21 Jan 11 2020 20:44:10
%S A322784 1,1,4,8,29,59,311,892,4983,21863,126813,678626,4446565,27644538,
%T A322784 195561593,1384705697,10613378402,82864870101,686673571479,
%U A322784 5832742205547,51897707277698,474889512098459,4514467567213008,44152005855085601,446355422070799305,4638590359349994120
%N A322784 Number of multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.
%C A322784 A multiset is uniform if all multiplicities are equal.
%C A322784 Also the number of factorizations into factors > 1 of primorial powers k in A100778 with sum of prime indices A056239(k) equal to n.
%C A322784 a(n) is the number of nonequivalent nonnegative integer matrices without zero rows or columns with equal column sums and total sum n up to permutation of rows. - _Andrew Howroyd_, Jan 11 2020
%H A322784 Andrew Howroyd, <a href="/A322784/b322784.txt">Table of n, a(n) for n = 0..50</a>
%F A322784 a(n) = Sum_{d|n} A001055(A002110(n/d)^d).
%F A322784 a(n) = Sum_{d|n} A219727(n/d, d). - _Andrew Howroyd_, Jan 11 2020
%e A322784 The a(1) = 1 through a(4) = 29 multiset partitions:
%e A322784   {{1}}   {{1,1}}     {{1,1,1}}       {{1,1,1,1}}
%e A322784           {{1,2}}     {{1,2,3}}       {{1,1,2,2}}
%e A322784          {{1},{1}}   {{1},{1,1}}      {{1,2,3,4}}
%e A322784          {{1},{2}}   {{1},{2,3}}     {{1},{1,1,1}}
%e A322784                      {{2},{1,3}}     {{1,1},{1,1}}
%e A322784                      {{3},{1,2}}     {{1},{1,2,2}}
%e A322784                     {{1},{1},{1}}    {{1,1},{2,2}}
%e A322784                     {{1},{2},{3}}    {{1,2},{1,2}}
%e A322784                                      {{1},{2,3,4}}
%e A322784                                      {{1,2},{3,4}}
%e A322784                                      {{1,3},{2,4}}
%e A322784                                      {{1,4},{2,3}}
%e A322784                                      {{2},{1,1,2}}
%e A322784                                      {{2},{1,3,4}}
%e A322784                                      {{3},{1,2,4}}
%e A322784                                      {{4},{1,2,3}}
%e A322784                                     {{1},{1},{1,1}}
%e A322784                                     {{1},{1},{2,2}}
%e A322784                                     {{1},{2},{1,2}}
%e A322784                                     {{1},{2},{3,4}}
%e A322784                                     {{1},{3},{2,4}}
%e A322784                                     {{1},{4},{2,3}}
%e A322784                                     {{2},{2},{1,1}}
%e A322784                                     {{2},{3},{1,4}}
%e A322784                                     {{2},{4},{1,3}}
%e A322784                                     {{3},{4},{1,2}}
%e A322784                                    {{1},{1},{1},{1}}
%e A322784                                    {{1},{1},{2},{2}}
%e A322784                                    {{1},{2},{3},{4}}
%t A322784 u[n_,k_]:=u[n,k]=If[n==1,1,Sum[u[n/d,d],{d,Select[Rest[Divisors[n]],#<=k&]}]];
%t A322784 Table[Sum[u[Array[Prime,d,1,Times]^(n/d),Array[Prime,d,1,Times]^(n/d)],{d,Divisors[n]}],{n,12}]
%Y A322784 Row sums of A322786.
%Y A322784 Cf. A001055, A005176, A056239, A072774, A100778, A219727, A295193, A306017 (unlabeled version), A319190, A319612, A322785, A322786, A322792.
%K A322784 nonn
%O A322784 0,3
%A A322784 _Gus Wiseman_, Dec 26 2018
%E A322784 a(14)-a(15) from _Alois P. Heinz_, Jan 16 2019
%E A322784 Terms a(16) and beyond from _Andrew Howroyd_, Jan 11 2020