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A237978
Number of partitions of n having depth 3; see Comments.
3
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 1, 3, 1, 10, 6, 6, 13, 15, 20, 30, 30, 64, 61, 85, 99, 153, 170, 184, 271, 359, 416, 539, 631, 799, 971, 1199, 1433, 1803, 2153, 2692, 3173, 3882, 4544, 5610, 6663, 8090, 9565, 11467, 13590, 16305, 19184, 22933, 26973
OFFSET
1,12
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. Conjecture: if d >= 0, then 2^d is the least n that has a partition of depth d.
EXAMPLE
a(14) = 3 counts these partitions: 64211, 632111, 433211.
Successive applications of f to the first of these partitions are indicated by 64211 -> 6422 -> 644 -> 86.
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}] (* this sequence *)
(* Peter J. C. Moses, Feb 19 2014 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 19 2014
STATUS
approved