

A238003


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


1



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
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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(n1,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(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



