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%I #24 Sep 30 2023 21:54:15
%S 0,1,1,2,4,6,9,11,20,25,37,47,67,85,122,142,200,259,330,412,538,663,
%T 846,1026,1309,1598,2013,2432,3003,3670,4467,5383,6591,7892,9544,
%U 11472,13768,16424,19686,23392,27802,33011,39094,46243,54700,64273,75638,88765
%N Number of partitions of n having depth 1; see Comments.
%C 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 The 11 partitions of 6 are partitioned by depth as follows:
%e depth 0: 6, 51, 42, 321;
%e depth 1: 411, 33, 222, 2211, 21111, 11111;
%e depth 2: 3111.
%e Thus, a(6) = 6, A000009(6) = 4, A237750(6) = 1, A237978(6) = 0.
%t z = 60; c[n_] := c[n] = Map[Length[FixedPointList[Sort[Map[Total, Split[#]], Greater] &, #]] - 2 &, IntegerPartitions[n]]
%t Table[Count[c[n], 1], {n, 1, z}] (* this sequence *)
%t Table[Count[c[n], 2], {n, 1, z}] (* A237750 *)
%t Table[Count[c[n], 3], {n, 1, z}] (* A237978 *)
%t (* _Peter J. C. Moses_, Feb 19 2014 *)
%Y Cf. A237750, A237978, A366063, A000009, A000041.
%K nonn,easy
%O 1,4
%A _Clark Kimberling_, Feb 19 2014
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