A237978 - OEIS

%I #14 Sep 30 2023 21:54:35

%S 0,0,0,0,0,0,0,1,0,0,1,2,1,3,1,10,6,6,13,15,20,30,30,64,61,85,99,153,

%T 170,184,271,359,416,539,631,799,971,1199,1433,1803,2153,2692,3173,

%U 3882,4544,5610,6663,8090,9565,11467,13590,16305,19184,22933,26973

%N Number of partitions of n having depth 3; 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 a(14) = 3 counts these partitions: 64211, 632111, 433211.

%e Successive applications of f to the first of these partitions are indicated by 64211 -> 6422 -> 644 -> 86.

%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}] (* A237685 *)

%t Table[Count[c[n], 2], {n, 1, z}] (* A237750 *)

%t Table[Count[c[n], 3], {n, 1, z}] (* this sequence *)

%t (* _Peter J. C. Moses_, Feb 19 2014 *)

%Y Cf. A237685, A237750, A366063, A000009, A000041.

%K nonn,easy

%O 1,12

%A _Clark Kimberling_, Feb 19 2014