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

1, 1, 1, 4, 6, 11, 28, 45, 86, 172, 344, 608, 1135, 2206, 4006, 7689, 13748, 25502, 47406, 86838, 157560, 286642, 522089, 941356, 1718622, 3079218, 5525805, 9902996, 17788396, 31742616, 56694704, 100720516, 178468026, 317019140, 560079704, 991061957

Offset: 0

Author

Gus Wiseman, Dec 11 2022

Keywords

• nonn

Examples

			The a(1) = 1 through a(5) = 11 sequences of sets:
  ({1})  ({2})  ({3})      ({4})        ({5})
                ({1,2})    ({1,3})      ({1,4})
                ({1},{2})  ({1},{3})    ({2,3})
                ({2},{1})  ({3},{1})    ({1},{4})
                           ({1},{1,2})  ({2},{3})
                           ({1,2},{1})  ({3},{2})
                                        ({4},{1})
                                        ({1},{1,3})
                                        ({1,2},{2})
                                        ({1,3},{1})
                                        ({2},{1,2})
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

The unordered version is A050342, non-strict A261049.

The case of strictly decreasing sums is A279785.

This is the distinct case of A304969.

The case of distinct sums is A336343, constant sums A279791.

This is the case of A358906 with strict partitions.

The version for compositions instead of strict partitions is A358907.

The case of twice-partitions is A358914.

A001970 counts multiset partitions of integer partitions.

A055887 counts sequences of partitions.

A063834 counts twice-partitions.

A330462 counts set systems by total sum and length.

A358830 counts twice-partitions with distinct lengths.

Cf. A000009, A000041, A000219, A271619, A296122, A336342, A358908.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(binomial(g(i), j)*b(n-i*j, i-1, p+j), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 13 2024

Mathematica

ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n],UnsameQ@@#&&And@@UnsameQ@@@#&]],{n,0,10}]

Formula

a(n) = Sum_{k} A330462(n,k) * k!.