A237978 Number of partitions of n having depth 3; see Comments.

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, 170, 184, 271, 359, 416, 539, 631, 799, 971, 1199, 1433, 1803, 2153, 2692, 3173, 3882, 4544, 5610, 6663, 8090, 9565, 11467, 13590, 16305, 19184, 22933, 26973

Offset

1,12

Comments

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.

Links

Table of n, a(n) for n=1..55.

Example

a(14) = 3 counts these partitions: 64211, 632111, 433211.
Successive applications of f to the first of these partitions are indicated by 64211 -> 6422 -> 644 -> 86.

Mathematica

z = 60; c[n_] := c[n] = Map[Length[FixedPointList[Sort[Map[Total, Split[#]], Greater] &, #]] - 2 &, IntegerPartitions[n]]
Table[Count[c[n], 1], {n, 1, z}] (* A237685 *)
Table[Count[c[n], 2], {n, 1, z}] (* A237750 *)
Table[Count[c[n], 3], {n, 1, z}] (* this sequence *)
(* Peter J. C. Moses, Feb 19 2014 *)

Crossrefs

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

Sequence in context: A088640 A186725 A301935 * A337276 A098570 A122048

Adjacent sequences: A237975 A237976 A237977 * A237979 A237980 A237981

Keyword

nonn,easy

Author

Clark Kimberling, Feb 19 2014

Status

approved