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 A237685 Number of partitions of n having depth 1; see Comments. 5
 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:  the least n that has depth d is 2^n. LINKS 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. A237750, A237978, 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 January 29 16:58 EST 2020. Contains 331347 sequences. (Running on oeis4.)