A358914 Number of twice-partitions of n into distinct strict partitions.

1, 1, 1, 3, 4, 7, 13, 20, 32, 51, 83, 130, 206, 320, 496, 759, 1171, 1786, 2714, 4104, 6193, 9286, 13920, 20737, 30865, 45721, 67632, 99683, 146604, 214865, 314782, 459136, 668867, 972425, 1410458, 2040894, 2950839, 4253713, 6123836, 8801349, 12627079

Offset: 0

Author

Gus Wiseman, Dec 11 2022

Keywords

• nonn

Comments

A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.

Examples

			The a(1) = 1 through a(6) = 13 twice-partitions:
  ((1))  ((2))  ((3))     ((4))      ((5))      ((6))
                ((21))    ((31))     ((32))     ((42))
                ((2)(1))  ((3)(1))   ((41))     ((51))
                          ((21)(1))  ((3)(2))   ((321))
                                     ((4)(1))   ((4)(2))
                                     ((21)(2))  ((5)(1))
                                     ((31)(1))  ((21)(3))
                                                ((31)(2))
                                                ((3)(21))
                                                ((32)(1))
                                                ((41)(1))
                                                ((3)(2)(1))
                                                ((21)(2)(1))
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..100

Crossrefs

The unordered version is A050342, non-strict A261049.

This is the distinct case of A270995.

The case of strictly decreasing sums is A279785.

The case of constant sums is A279791.

For distinct instead of weakly decreasing sums we have A336343.

This is the twice-partition case of A358913.

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, A000219, A075900, A271619, A296122, A304969, A321449, A336342, A358901, A358906, A358907.

Programs

Mathematica

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

PARI

seq(n,k)={my(u=Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))-1)); Vec(prod(k=1, n, my(c=u[k]); sum(j=0, min(c,n\k), x^(j*k)*c!/(c-j)!,  O(x*x^n))))} \\ Andrew Howroyd, Dec 31 2022

Extensions

Terms a(26) and beyond from Andrew Howroyd, Dec 31 2022