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 A237685 Number of partitions of n having depth 1; see Comments. 20
 0, 1, 1, 2, 4, 6, 9, 11, 20, 25, 37, 47, 67, 85, 122, 142, 200, 259, 330, 412, 538, 663, 846, 1026, 1309, 1598, 2013, 2432, 3003, 3670, 4467, 5383, 6591, 7892, 9544, 11472, 13768, 16424, 19686, 23392, 27802, 33011, 39094, 46243, 54700, 64273, 75638, 88765 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 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. LINKS Table of n, a(n) for n=1..48. 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}] (* this sequence *) 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. A237750, A237978, A366063, A000009, A000041. Sequence in context: A195526 A153196 A247185 * A220768 A077220 A128716 Adjacent sequences: A237682 A237683 A237684 * A237686 A237687 A237688 KEYWORD nonn,easy AUTHOR Clark Kimberling, Feb 19 2014 STATUS approved

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Last modified July 25 08:50 EDT 2024. Contains 374587 sequences. (Running on oeis4.)