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Number of sequences of integer partitions with total sum n that are rectangular, meaning all lengths are equal.
10

%I #10 Dec 31 2022 11:20:07

%S 1,1,3,6,13,24,49,91,179,341,664,1280,2503,4872,9557,18750,36927,

%T 72800,143880,284660,564093,1118911,2221834,4415417,8781591,17476099,

%U 34799199,69327512,138176461,275503854,549502119,1096327380,2187894634,4367310138,8719509111

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

%H Andrew Howroyd, <a href="/A358905/b358905.txt">Table of n, a(n) for n = 0..1000</a>

%F 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

%e The a(0) = 1 through a(4) = 13 sequences:

%e () ((1)) ((2)) ((3)) ((4))

%e ((11)) ((21)) ((22))

%e ((1)(1)) ((111)) ((31))

%e ((1)(2)) ((211))

%e ((2)(1)) ((1111))

%e ((1)(1)(1)) ((1)(3))

%e ((2)(2))

%e ((3)(1))

%e ((11)(11))

%e ((1)(1)(2))

%e ((1)(2)(1))

%e ((2)(1)(1))

%e ((1)(1)(1)(1))

%t ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];

%t Table[Length[Select[ptnseq[n],SameQ@@Length/@#&]],{n,0,10}]

%o (PARI)

%o P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}

%o 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

%Y The case of set partitions is A038041.

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

%Y For equal sums instead of lengths we have A279787.

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

%Y The unordered version is A319066.

%Y The case of plane partitions is A323429.

%Y The case of constant sums also is A358833.

%Y A055887 counts sequences of partitions with total sum n.

%Y A281145 counts same-trees.

%Y A319169 counts partitions with constant Omega, ranked by A320324.

%Y A358911 counts compositions with constant Omega, distinct A358912.

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 07 2022

%E Terms a(16) and beyond from _Andrew Howroyd_, Dec 31 2022