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

0, 0, 0, 1, 0, 1, 1, 4, 2, 7, 6, 13, 15, 25, 26, 46, 53, 74, 92, 136, 157, 218, 274, 356, 443, 583, 703, 899, 1125, 1447, 1746, 2182, 2661, 3331, 4077, 4997, 6066, 7432, 8984, 10904, 13212, 15845, 19161, 22932, 27526, 32968, 39351, 46778, 55791, 66272, 78480

Offset

1,8

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..51.

Example

The 11 partitions of 6 are partitioned by depth as follows:
  depth 0: 6, 51, 42, 321;
  depth 1: 411, 33, 222, 2211, 21111, 11111;
  depth 2: 3111.
Thus, a(6) = 6, A000009(6) = 4, A237750(6) = 1, A237978(6) = 0.

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}] (* this sequence *)
Table[Count[c[n], 3], {n, 1, z}] (* A237978 *)
(* Peter J. C. Moses, Feb 19 2014 *)

Crossrefs

Cf. A237685, A237978, A366063, A000009, A000041.

Sequence in context: A125271 A245262 A092314 * A249652 A110841 A128226

Adjacent sequences: A237747 A237748 A237749 * A237751 A237752 A237753

Keyword

nonn,easy

Author

Clark Kimberling, Feb 19 2014

Status

approved