cp's OEIS Frontend

A237750 Number of partitions of n having depth 2; see Comments.

%I A237750 #22 Feb 25 2025 04:49:51
%S A237750 0,0,0,1,0,1,1,4,2,7,6,13,15,25,26,46,53,74,92,136,157,218,274,356,
%T A237750 443,583,703,899,1125,1447,1746,2182,2661,3331,4077,4997,6066,7432,
%U A237750 8984,10904,13212,15845,19161,22932,27526,32968,39351,46778,55791,66272,78480
%N A237750 Number of partitions of n having depth 2; see Comments.
%C A237750 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 A237750 The 11 partitions of 6 are partitioned by depth as follows:
%e A237750   depth 0: 6, 51, 42, 321;
%e A237750   depth 1: 411, 33, 222, 2211, 21111, 11111;
%e A237750   depth 2: 3111.
%e A237750 Thus, a(6) = 6, A000009(6) = 4, A237750(6) = 1, A237978(6) = 0.
%t A237750 z = 60; c[n_] := c[n] = Map[Length[FixedPointList[Sort[Map[Total, Split[#]], Greater] &, #]] - 2 &, IntegerPartitions[n]]
%t A237750 Table[Count[c[n], 1], {n, 1, z}] (* A237685 *)
%t A237750 Table[Count[c[n], 2], {n, 1, z}] (* this sequence *)
%t A237750 Table[Count[c[n], 3], {n, 1, z}] (* A237978 *)
%t A237750 (* _Peter J. C. Moses_, Feb 19 2014 *)
%Y A237750 Cf. A237685, A237978, A366063, A000009, A000041.
%K A237750 nonn,easy
%O A237750 1,8
%A A237750 _Clark Kimberling_, Feb 19 2014