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

1, 1, 2, 3, 3, 5, 6, 11, 10, 17, 19, 30, 34, 50, 54, 89, 97, 126, 160, 215, 254, 339, 409, 549, 649, 838, 997, 1286, 1562, 1934, 2375, 2966, 3552, 4418, 5339, 6505, 7869, 9591, 11499, 13946, 16781, 20163, 24167, 28932, 34434, 41285, 49116, 58508, 69361

Offset: 1

Author

Clark Kimberling, Feb 19 2014

Keywords

• nonn
• easy

Comments

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.

Examples

			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) = 5.
		

Crossrefs

Cf. A237685, A000041.

Programs

Mathematica

z = 40; 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[PartitionsP[n] - Count[c[n], 1], {n, 1, z}]  (* A238003 *)
(* Peter J. C. Moses, Feb 19 2014 *)

Formula

a(n) = A000041(n) - A237685(n) for n >= 1.