

A237750


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


3



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
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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(n1,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.


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}] (* A237750 *)
Table[Count[c[n], 3], {n, 1, z}] (* A237978 *)
(* Peter J. C. Moses, Feb 19 2014 *)


CROSSREFS

Cf. A237685, A237978, 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



