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

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(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)
		

Links

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

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.

Programs

Mathematica

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

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(jAndrew Howroyd, Dec 31 2022

Extensions

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