A358832 Number of twice-partitions of n into partitions of distinct lengths and distinct sums.

1, 1, 2, 4, 7, 15, 25, 49, 79, 154, 248, 453, 748, 1305, 2125, 3702, 5931, 9990, 16415, 26844, 43246, 70947, 113653, 182314, 292897, 464614, 739640, 1169981, 1844511, 2888427, 4562850, 7079798, 11064182, 17158151, 26676385, 41075556, 63598025, 97420873, 150043132

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)
                      (21)(1)   (2111)
                      (111)(1)  (11111)
                                (21)(2)
                                (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@@Total/@#&&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-1, 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

This is the case of A271619 with distinct lengths.

These multiset partitions are ranked by A326535 /\ A326533.

This is the case of A358830 with distinct sums.

For constant instead of distinct lengths and sums we have A358833.

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

A273873 counts strict trees.

Cf. A000009, A000219, A001970, A141199, A279375, A279785, A279790, A306319, A358334, A358836.

Sequence in context: A171276 A027167 A259090 * A232464 A264292 A259592

Adjacent sequences: A358829 A358830 A358831 * A358833 A358834 A358835

Keyword

nonn

Author

Gus Wiseman, Dec 04 2022

Extensions

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2022

Status

approved