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

1,3

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.

Links

Table of n, a(n) for n=1..49.

Formula

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

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) = 5.

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 *)

Crossrefs

Cf. A237685, A000041.

Sequence in context: A241409 A018131 A121400 * A218932 A056878 A270520

Adjacent sequences: A238000 A238001 A238002 * A238004 A238005 A238006

Keyword

nonn,easy

Author

Clark Kimberling, Feb 19 2014

Status

approved