cp's OEIS Frontend

A353698 Number of integer partitions of n whose product equals their length.

%I A353698 #10 Jan 02 2023 21:54:42
%S A353698 0,1,0,1,0,1,1,1,0,2,0,2,1,2,0,2,1,2,1,1,1,4,1,2,1,2,1,3,0,3,2,2,1,5,
%T A353698 0,1,2,5,1,4,0,3,3,2,1,4,2,3,2,2,0,5,1,4,2,2,3,6,1,2,2,5,1,4,0,4,3,3,
%U A353698 1,6,2,3,4,4,2,4,1,4,2,3,1,8,2,4,2,4,2,5,2,4,2
%N A353698 Number of integer partitions of n whose product equals their length.
%H A353698 Andrew Howroyd, <a href="/A353698/b353698.txt">Table of n, a(n) for n = 0..10000</a>
%e A353698 The a(n) partitions for selected n (A..H = 10..17):
%e A353698 n=9:    n=21:             n=27:                 n=33:
%e A353698 ---------------------------------------------------------------------------
%e A353698 51111   B1111111111       E1111111111111        H1111111111111111
%e A353698 321111  72111111111111    921111111111111111    B211111111111111111111
%e A353698         531111111111111   54111111111111111111  831111111111111111111111
%e A353698         4221111111111111                        5511111111111111111111111
%e A353698                                                 333111111111111111111111111
%t A353698 Table[Length[Select[IntegerPartitions[n],Times@@#==Length[#]&]],{n,0,30}]
%o A353698 (PARI) a(r,m=r,p=1,k=0) = {(p==k+r) + sum(m=2, min(m, (k+r)\p),  self()(r-m, min(m,r-m), p*m, k+1))} \\ _Andrew Howroyd_, Jan 02 2023
%Y A353698 The LHS (product of parts) is counted by A339095, rank statistic A003963.
%Y A353698 The RHS (length) is counted by A008284, rank statistic A001222.
%Y A353698 These partitions are ranked by A353699.
%Y A353698 A266477 counts partitions by product of multiplicities, rank stat A005361.
%Y A353698 A353504 counts partitions w/ product less than product of multiplicities.
%Y A353698 A353505 counts partitions w/ product greater than product of multiplicities.
%Y A353698 A353506 counts partitions w/ prod equal to prod of mults, ranked by A353503.
%Y A353698 Cf. A000041, A002033, A098859, A114640, A181819, A225485, A266499, A319000, A325280, A353398, A353507.
%K A353698 nonn
%O A353698 0,10
%A A353698 _Gus Wiseman_, May 19 2022
%E A353698 Terms a(61) and beyond from _Andrew Howroyd_, Jan 02 2023