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

0,3

Links

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

Formula

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

Example

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))

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}]

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.

Sequence in context: A051748 A330452 A086904 * A360869 A291276 A358274

Adjacent sequences: A358903 A358904 A358905 * A358907 A358908 A358909

Keyword

nonn

Author

Gus Wiseman, Dec 07 2022

Status

approved