cp's OEIS Frontend

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

%I A358905 #10 Dec 31 2022 11:20:07
%S A358905 1,1,3,6,13,24,49,91,179,341,664,1280,2503,4872,9557,18750,36927,
%T A358905 72800,143880,284660,564093,1118911,2221834,4415417,8781591,17476099,
%U A358905 34799199,69327512,138176461,275503854,549502119,1096327380,2187894634,4367310138,8719509111
%N A358905 Number of sequences of integer partitions with total sum n that are rectangular, meaning all lengths are equal.
%H A358905 Andrew Howroyd, <a href="/A358905/b358905.txt">Table of n, a(n) for n = 0..1000</a>
%F A358905 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 A358905 The a(0) = 1 through a(4) = 13 sequences:
%e A358905   ()  ((1))  ((2))     ((3))        ((4))
%e A358905              ((11))    ((21))       ((22))
%e A358905              ((1)(1))  ((111))      ((31))
%e A358905                        ((1)(2))     ((211))
%e A358905                        ((2)(1))     ((1111))
%e A358905                        ((1)(1)(1))  ((1)(3))
%e A358905                                     ((2)(2))
%e A358905                                     ((3)(1))
%e A358905                                     ((11)(11))
%e A358905                                     ((1)(1)(2))
%e A358905                                     ((1)(2)(1))
%e A358905                                     ((2)(1)(1))
%e A358905                                     ((1)(1)(1)(1))
%t A358905 ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
%t A358905 Table[Length[Select[ptnseq[n],SameQ@@Length/@#&]],{n,0,10}]
%o A358905 (PARI)
%o A358905 P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
%o A358905 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 A358905 The case of set partitions is A038041.
%Y A358905 The version for weakly decreasing lengths is A141199, strictly A358836.
%Y A358905 For equal sums instead of lengths we have A279787.
%Y A358905 The case of twice-partitions is A306319, distinct A358830.
%Y A358905 The unordered version is A319066.
%Y A358905 The case of plane partitions is A323429.
%Y A358905 The case of constant sums also is A358833.
%Y A358905 A055887 counts sequences of partitions with total sum n.
%Y A358905 A281145 counts same-trees.
%Y A358905 A319169 counts partitions with constant Omega, ranked by A320324.
%Y A358905 A358911 counts compositions with constant Omega, distinct A358912.
%Y A358905 Cf. A000041, A000219, A001970, A063834, A218482, A305551, A358835.
%K A358905 nonn
%O A358905 0,3
%A A358905 _Gus Wiseman_, Dec 07 2022
%E A358905 Terms a(16) and beyond from _Andrew Howroyd_, Dec 31 2022