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

Author

Gus Wiseman, Dec 04 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)
                      (21)(1)   (2111)
                      (111)(1)  (11111)
                                (21)(2)
                                (22)(1)
                                (3)(11)
                                (31)(1)
                                (111)(2)
                                (211)(1)
                                (111)(11)
                                (1111)(1)
		

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.

Programs

Mathematica

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],UnsameQ@@Total/@#&&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-1,r-m), r-m, bitor(b, 1<Andrew Howroyd, Dec 31 2022

Extensions

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