cp's OEIS Frontend

A358913 Number of finite sequences of distinct sets with total sum n.

%I A358913 #10 Feb 13 2024 20:09:46
%S A358913 1,1,1,4,6,11,28,45,86,172,344,608,1135,2206,4006,7689,13748,25502,
%T A358913 47406,86838,157560,286642,522089,941356,1718622,3079218,5525805,
%U A358913 9902996,17788396,31742616,56694704,100720516,178468026,317019140,560079704,991061957
%N A358913 Number of finite sequences of distinct sets with total sum n.
%H A358913 Alois P. Heinz, <a href="/A358913/b358913.txt">Table of n, a(n) for n = 0..1000</a>
%F A358913 a(n) = Sum_{k} A330462(n,k) * k!.
%e A358913 The a(1) = 1 through a(5) = 11 sequences of sets:
%e A358913   ({1})  ({2})  ({3})      ({4})        ({5})
%e A358913                 ({1,2})    ({1,3})      ({1,4})
%e A358913                 ({1},{2})  ({1},{3})    ({2,3})
%e A358913                 ({2},{1})  ({3},{1})    ({1},{4})
%e A358913                            ({1},{1,2})  ({2},{3})
%e A358913                            ({1,2},{1})  ({3},{2})
%e A358913                                         ({4},{1})
%e A358913                                         ({1},{1,3})
%e A358913                                         ({1,2},{2})
%e A358913                                         ({1,3},{1})
%e A358913                                         ({2},{1,2})
%p A358913 g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
%p A358913      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A358913     end:
%p A358913 b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
%p A358913       add(binomial(g(i), j)*b(n-i*j, i-1, p+j), j=0..n/i)))
%p A358913     end:
%p A358913 a:= n-> b(n$2, 0):
%p A358913 seq(a(n), n=0..35);  # _Alois P. Heinz_, Feb 13 2024
%t A358913 ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
%t A358913 Table[Length[Select[ptnseq[n],UnsameQ@@#&&And@@UnsameQ@@@#&]],{n,0,10}]
%Y A358913 The unordered version is A050342, non-strict A261049.
%Y A358913 The case of strictly decreasing sums is A279785.
%Y A358913 This is the distinct case of A304969.
%Y A358913 The case of distinct sums is A336343, constant sums A279791.
%Y A358913 This is the case of A358906 with strict partitions.
%Y A358913 The version for compositions instead of strict partitions is A358907.
%Y A358913 The case of twice-partitions is A358914.
%Y A358913 A001970 counts multiset partitions of integer partitions.
%Y A358913 A055887 counts sequences of partitions.
%Y A358913 A063834 counts twice-partitions.
%Y A358913 A330462 counts set systems by total sum and length.
%Y A358913 A358830 counts twice-partitions with distinct lengths.
%Y A358913 Cf. A000009, A000041, A000219, A271619, A296122, A336342, A358908.
%K A358913 nonn
%O A358913 0,4
%A A358913 _Gus Wiseman_, Dec 11 2022