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

0,3

Comments

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

Links

Table of n, a(n) for n=0..38.

Example

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)

Mathematica

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

Prog

(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<<k)))));
  mapput(Cache, key, z)); z);
  vector(n+1, i, F(i-1, i-1, 0))
} \\ Andrew Howroyd, Dec 31 2022

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.

Sequence in context: A083270 A000879 A262232 * A321410 A218912 A230868

Adjacent sequences: A358827 A358828 A358829 * A358831 A358832 A358833

Keyword

nonn

Author

Gus Wiseman, Dec 03 2022

Extensions

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

Status

approved