A358906 Number of finite sequences of distinct integer partitions with total sum n.

1, 1, 2, 7, 13, 35, 87, 191, 470, 1080, 2532, 5778, 13569, 30715, 69583, 160386, 360709, 814597, 1824055, 4102430, 9158405, 20378692, 45215496, 100055269, 221388993, 486872610, 1069846372, 2343798452, 5127889666, 11186214519, 24351106180, 52896439646

Offset: 0

Author

Gus Wiseman, Dec 07 2022

Keywords

• nonn

Examples

			The a(1) = 1 through a(4) = 13 sequences:
  ((1))  ((2))   ((3))      ((4))
         ((11))  ((21))     ((22))
                 ((111))    ((31))
                 ((1)(2))   ((211))
                 ((2)(1))   ((1111))
                 ((1)(11))  ((1)(3))
                 ((11)(1))  ((3)(1))
                            ((11)(2))
                            ((1)(21))
                            ((2)(11))
                            ((21)(1))
                            ((1)(111))
                            ((111)(1))
		

Links

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

Crossrefs

This is the case of A055887 with distinct partitions.

The unordered version is A261049.

The case of twice-partitions is A296122.

The case of distinct sums is A336342, constant sums A279787.

The version for sequences of compositions is A358907.

The case of weakly decreasing lengths is A358908.

The case of distinct lengths is A358912.

The version for strict partitions is A358913, distinct case of A304969.

A001970 counts multiset partitions of integer partitions.

A063834 counts twice-partitions.

A358830 counts twice-partitions with distinct lengths.

A358901 counts partitions with all distinct Omegas.

Cf. A000009, A000041, A000219, A098407, A271619, A330463, A358836, A358901, A358914.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      binomial(combinat[numbpart](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..32);  # 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@@#&]],{n,0,10}]

Formula

a(n) = Sum_{k} A330463(n,k) * k!.