A358907 Number of finite sequences of distinct integer compositions with total sum n.

1, 1, 2, 8, 18, 54, 156, 412, 1168, 3200, 8848, 24192, 66632, 181912, 495536, 1354880, 3680352, 9997056, 27093216, 73376512, 198355840, 535319168, 1443042688, 3884515008, 10445579840, 28046885824, 75225974912, 201536064896, 539339293824, 1441781213952

Offset: 0

Author

Gus Wiseman, Dec 07 2022

Keywords

• nonn

Examples

			The a(1) = 1 through a(4) = 18 sequences:
  ((1))  ((2))   ((3))      ((4))
         ((11))  ((12))     ((13))
                 ((21))     ((22))
                 ((111))    ((31))
                 ((1)(2))   ((112))
                 ((2)(1))   ((121))
                 ((1)(11))  ((211))
                 ((11)(1))  ((1111))
                            ((1)(3))
                            ((3)(1))
                            ((1)(12))
                            ((11)(2))
                            ((1)(21))
                            ((12)(1))
                            ((2)(11))
                            ((21)(1))
                            ((1)(111))
                            ((111)(1))
		

Links

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

Crossrefs

For sets instead of sequences we have A098407, partitions A261049.

This is the strict case of A133494.

The case of distinct sums is A336127, constant sums A074854.

The version for sequences of partitions is A358906.

A001970 counts multiset partitions of integer partitions.

A063834 counts twice-partitions.

A218482 counts sequences of compositions with weakly decreasing lengths.

A358830 counts twice-partitions with distinct lengths.

A358901 counts partitions with all different Omegas.

A358914 counts twice-partitions into distinct strict partitions.

Cf. A000009, A000041, A000219, A055887, A075900, A296122, A304961, A307068, A336342, A358836, A358912.

Programs

Maple

g:= proc(n) option remember; ceil(2^(n-1)) end:
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, (t->
      add(binomial(t, j)*b(n-i*j, i-1, p+j), j=0..min(t, n/i)))(g(i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..32);  # Alois P. Heinz, Dec 15 2022

Mathematica

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

Extensions

a(16)-a(29) from Alois P. Heinz, Dec 15 2022