A358905 Number of sequences of integer partitions with total sum n that are rectangular, meaning all lengths are equal.

1, 1, 3, 6, 13, 24, 49, 91, 179, 341, 664, 1280, 2503, 4872, 9557, 18750, 36927, 72800, 143880, 284660, 564093, 1118911, 2221834, 4415417, 8781591, 17476099, 34799199, 69327512, 138176461, 275503854, 549502119, 1096327380, 2187894634, 4367310138, 8719509111

Offset: 0

Author

Gus Wiseman, Dec 07 2022

Keywords

• nonn

Examples

			The a(0) = 1 through a(4) = 13 sequences:
  ()  ((1))  ((2))     ((3))        ((4))
             ((11))    ((21))       ((22))
             ((1)(1))  ((111))      ((31))
                       ((1)(2))     ((211))
                       ((2)(1))     ((1111))
                       ((1)(1)(1))  ((1)(3))
                                    ((2)(2))
                                    ((3)(1))
                                    ((11)(11))
                                    ((1)(1)(2))
                                    ((1)(2)(1))
                                    ((2)(1)(1))
                                    ((1)(1)(1)(1))
		

Links

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

Crossrefs

The case of set partitions is A038041.

The version for weakly decreasing lengths is A141199, strictly A358836.

For equal sums instead of lengths we have A279787.

The case of twice-partitions is A306319, distinct A358830.

The unordered version is A319066.

The case of plane partitions is A323429.

The case of constant sums also is A358833.

A055887 counts sequences of partitions with total sum n.

A281145 counts same-trees.

A319169 counts partitions with constant Omega, ranked by A320324.

A358911 counts compositions with constant Omega, distinct A358912.

Cf. A000041, A000219, A001970, A063834, A218482, A305551, A358835.

Programs

Mathematica

ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n],SameQ@@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=P(n,y)); Vec(1 + sum(k=1, n, 1/(1 - polcoef(g, k, y)) - 1))} \\ Andrew Howroyd, Dec 31 2022

Formula

G.f.: 1 + Sum_{k>=1} (1/(1 - [y^k]P(x,y)) - 1) where P(x,y) = 1/Product_{k>=1} (1 - y*x^k). - Andrew Howroyd, Dec 31 2022

Extensions

Terms a(16) and beyond from Andrew Howroyd, Dec 31 2022