A358830 Number of twice-partitions of n into partitions with all different lengths.

1, 1, 2, 4, 9, 15, 31, 53, 105, 178, 330, 555, 1024, 1693, 2991, 5014, 8651, 14242, 24477, 39864, 67078, 109499, 181311, 292764, 483775, 774414, 1260016, 2016427, 3254327, 5162407, 8285796, 13074804, 20812682, 32733603, 51717463, 80904644, 127305773, 198134675, 309677802

Offset: 0

Author

Gus Wiseman, Dec 03 2022

Keywords

• nonn

Comments

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

Examples

			The a(1) = 1 through a(5) = 15 twice-partitions:
  (1)  (2)   (3)      (4)       (5)
       (11)  (21)     (22)      (32)
             (111)    (31)      (41)
             (11)(1)  (211)     (221)
                      (1111)    (311)
                      (11)(2)   (2111)
                      (2)(11)   (11111)
                      (21)(1)   (21)(2)
                      (111)(1)  (22)(1)
                                (3)(11)
                                (31)(1)
                                (111)(2)
                                (211)(1)
                                (111)(11)
                                (1111)(1)
		

Crossrefs

The version for set partitions is A007837.

For sums instead of lengths we have A271619.

For constant instead of distinct lengths we have A306319.

The case of distinct sums also is A358832.

The version for multiset partitions of integer partitions is A358836.

A063834 counts twice-partitions, strict A296122, row-sums of A321449.

A273873 counts strict trees.

Cf. A000009, A000219, A001970, A141199, A279375, A279785, A279790, A336342, A358334, A358831.

Programs

Mathematica

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

PARI

seq(n)={ local(Cache=Map());
  my(g=Vec(-1+1/prod(k=1, n, 1 - y*x^k + O(x*x^n))));
  my(F(m,r,b) = my(key=[m,r,b], z); if(!mapisdefined(Cache,key,&z),
  z = if(r<=0||m==0, r==0, self()(m-1, r, b) + sum(k=1, m, my(c=polcoef(g[m],k)); if(!bittest(b,k)&&c, c*self()(min(m,r-m), r-m, bitor(b, 1<Andrew Howroyd, Dec 31 2022

Extensions

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