cp's OEIS Frontend

A238003 Number of partitions of n not having depth 1; see Comments.

%I A238003 #18 Feb 25 2025 20:44:11
%S A238003 1,1,2,3,3,5,6,11,10,17,19,30,34,50,54,89,97,126,160,215,254,339,409,
%T A238003 549,649,838,997,1286,1562,1934,2375,2966,3552,4418,5339,6505,7869,
%U A238003 9591,11499,13946,16781,20163,24167,28932,34434,41285,49116,58508,69361
%N A238003 Number of partitions of n not having depth 1; see Comments.
%C A238003 The depth of a partition is defined at A237685 as follows. 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 the depth of P. Conjecture: lim_{n->infinity} a(n)/A000041(n) = 1.
%F A238003 a(n) = A000041(n) - A237685(n) for n >= 1.
%e A238003 The 11 partitions of 6 are partitioned by depth as follows:
%e A238003   depth 0: 6, 51, 42, 321;
%e A238003   depth 1: 411, 33, 222, 2211, 21111, 11111;
%e A238003   depth 2: 3111.
%e A238003 Thus, a(6) = 5.
%t A238003 z = 40; c[n_] := c[n] = Map[Length[FixedPointList[Sort[Map[Total, Split[#]], Greater] &, #]] - 2 &, IntegerPartitions[n]]
%t A238003 Table[Count[c[n], 1], {n, 1, z}] (* A237685 *)
%t A238003 Table[PartitionsP[n] - Count[c[n], 1], {n, 1, z}]  (* A238003 *)
%t A238003 (* _Peter J. C. Moses_, Feb 19 2014 *)
%Y A238003 Cf. A237685, A000041.
%K A238003 nonn,easy
%O A238003 1,3
%A A238003 _Clark Kimberling_, Feb 19 2014