cp's OEIS Frontend

A237685 Number of partitions of n having depth 1; see Comments.

%I A237685 #28 Feb 25 2025 04:50:05
%S A237685 0,1,1,2,4,6,9,11,20,25,37,47,67,85,122,142,200,259,330,412,538,663,
%T A237685 846,1026,1309,1598,2013,2432,3003,3670,4467,5383,6591,7892,9544,
%U A237685 11472,13768,16424,19686,23392,27802,33011,39094,46243,54700,64273,75638,88765
%N A237685 Number of partitions of n having depth 1; see Comments.
%C A237685 Suppose that P is a partition of n. Let x(1), x(2),...,x(k) be the distinct parts of P, and let m(i) be the multiplicity of x(i) in P. Let f(P) be the partition [m(1)*x(1), m(2)*x(2),...,x(k)*m(k)] of n. Define c(0,P) = P, c(1,P) = f(P), ..., c(n,P) = f(c(n-1,P)), and define d(P) = least n such that c(n,P) has no repeated parts; d(P) is introduced here as the depth of P. Clearly d(P) = 0 if and only if P is a strict partition, as in A000009. Conjecture: if d >= 0, then 2^d is the least n that has a partition of depth d.
%e A237685 The 11 partitions of 6 are partitioned by depth as follows:
%e A237685   depth 0: 6, 51, 42, 321;
%e A237685   depth 1: 411, 33, 222, 2211, 21111, 11111;
%e A237685   depth 2: 3111.
%e A237685 Thus, a(6) = 6, A000009(6) = 4, A237750(6) = 1, A237978(6) = 0.
%t A237685 z = 60; c[n_] := c[n] = Map[Length[FixedPointList[Sort[Map[Total, Split[#]], Greater] &, #]] - 2 &, IntegerPartitions[n]]
%t A237685 Table[Count[c[n], 1], {n, 1, z}] (* this sequence *)
%t A237685 Table[Count[c[n], 2], {n, 1, z}] (* A237750 *)
%t A237685 Table[Count[c[n], 3], {n, 1, z}] (* A237978 *)
%t A237685 (* _Peter J. C. Moses_, Feb 19 2014 *)
%Y A237685 Cf. A237750, A237978, A366063, A000009, A000041.
%K A237685 nonn,easy
%O A237685 1,4
%A A237685 _Clark Kimberling_, Feb 19 2014