A358833 Number of rectangular twice-partitions of n of type (P,R,P).

1, 1, 3, 4, 8, 8, 17, 16, 32, 34, 56, 57, 119, 102, 179, 199, 335, 298, 598, 491, 960, 925, 1441, 1256, 2966, 2026, 3726, 3800, 6488, 4566, 11726, 6843, 16176, 14109, 21824, 16688, 49507, 21638, 50286, 50394, 99408, 44584, 165129, 63262, 208853, 205109, 248150

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, so these are twice-partitions of n into partitions with constant lengths and constant sums.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Gus Wiseman, Sequences enumerating triangles of integer partitions

Formula

a(n) = Sum_{d|n} Sum_{j=1..n/d} A008284(n/d, j)^d for n > 0. - Andrew Howroyd, Dec 31 2022

Example

The a(1) = 1 through a(5) = 8 twice-partitions:
  (1)  (2)     (3)        (4)           (5)
       (11)    (21)       (22)          (32)
       (1)(1)  (111)      (31)          (41)
               (1)(1)(1)  (211)         (221)
                          (1111)        (311)
                          (2)(2)        (2111)
                          (11)(11)      (11111)
                          (1)(1)(1)(1)  (1)(1)(1)(1)(1)

Mathematica

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

Prog

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

Crossrefs

This is the rectangular case of A279787.

This is the case of A306319 with constant sums.

For distinct instead of constant lengths and sums we have A358832.

The version for multiset partitions of integer partitions is A358835.

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

A281145 counts same-trees.

Cf. A000041, A000219, A001970, A008284, A141199, A327908, A358823, A358831.

Sequence in context: A030014 A047968 A358835 * A322117 A181778 A245026

Adjacent sequences: A358830 A358831 A358832 * A358834 A358835 A358836

Keyword

nonn

Author

Gus Wiseman, Dec 04 2022

Extensions

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

Status

approved