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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 (list; graph; refs; listen; history; text; internal format)
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.

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

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Last modified September 19 17:50 EDT 2020. Contains 337180 sequences. (Running on oeis4.)