A358831 Number of twice-partitions of n into partitions with weakly decreasing lengths.

1, 1, 3, 6, 14, 26, 56, 102, 205, 372, 708, 1260, 2345, 4100, 7388, 12819, 22603, 38658, 67108, 113465, 193876, 324980, 547640, 909044, 1516609, 2495023, 4118211, 6726997, 11002924, 17836022, 28948687, 46604803, 75074397, 120134298, 192188760, 305709858, 486140940

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

Andrew Howroyd, Table of n, a(n) for n = 0..500

Example

The a(1) = 1 through a(4) = 14 twice-partitions:
  (1)  (2)     (3)        (4)
       (11)    (21)       (22)
       (1)(1)  (111)      (31)
               (2)(1)     (211)
               (11)(1)    (1111)
               (1)(1)(1)  (2)(2)
                          (3)(1)
                          (11)(2)
                          (21)(1)
                          (11)(11)
                          (111)(1)
                          (2)(1)(1)
                          (11)(1)(1)
                          (1)(1)(1)(1)

Mathematica

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

Prog

(PARI)
P(n, y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=Vec(P(n, y)-1), v=[1]); for(k=1, n, my(p=g[k], u=v); v=vector(k+1); v[1] = 1 + O(x*x^n); for(j=1, k, v[1+j] = (v[j] + if(j<k, u[1+j] - u[j]))/(1 - polcoef(p, j)*x^k))); Vec(v[1+n])} \\ Andrew Howroyd, Dec 31 2022

Crossrefs

This is the semi-ordered case of A141199.

For constant instead of weakly decreasing lengths we have A306319.

For distinct instead of weakly decreasing lengths we have A358830.

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

A196545 counts p-trees, enriched A289501.

Cf. A000041, A000219, A001970, A061260, A072233, A271619, A279787, A358836.

Sequence in context: A120940 A049940 A265947 * A323450 A051749 A278788

Adjacent sequences: A358828 A358829 A358830 * A358832 A358833 A358834

Keyword

nonn

Author

Gus Wiseman, Dec 03 2022

Extensions

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

Status

approved