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 version for sequences of compositions is A358907.
The case of weakly decreasing lengths is A358908.
The case of distinct lengths is A358912.
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 07 2022
STATUS
approved