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A239955
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Number of partitions p of n such that (number of distinct parts of p) <= max(p) - min(p).
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12
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0, 0, 0, 0, 1, 2, 4, 7, 12, 17, 27, 38, 54, 75, 104, 137, 187, 245, 322, 418, 542, 691, 887, 1121, 1417, 1777, 2228, 2767, 3441, 4247, 5235, 6424, 7871, 9594, 11688, 14173, 17168, 20723, 24979, 30008, 36010, 43085, 51479, 61357, 73032, 86718, 102852, 121718
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OFFSET
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0,6
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COMMENTS
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Also the number of partitions of n with at least one gap, i.e., partitions whose parts do not form a contiguous interval. These partitions are ranked by A073492. For example, the a(0) = 0 through a(8) = 12 partitions are:
. . . . (31) (41) (42) (52) (53)
(311) (51) (61) (62)
(411) (331) (71)
(3111) (421) (422)
(511) (431)
(4111) (521)
(31111) (611)
(3311)
(4211)
(5111)
(41111)
(311111)
Also the number of non-constant partitions of n with a repeated non-maximal part, ranked by A065201. The a(0) = 0 through a(8) = 12 partitions are:
. . . . (211) (311) (411) (322) (422)
(2111) (2211) (511) (611)
(3111) (3211) (3221)
(21111) (4111) (3311)
(22111) (4211)
(31111) (5111)
(211111) (22211)
(32111)
(41111)
(221111)
(311111)
(2111111)
(End)
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LINKS
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FORMULA
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EXAMPLE
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a(6) counts these 4 partitions: 51, 42, 411, 3111.
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MATHEMATICA
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z = 60; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[p_] := f[p] = Max[p] - Min[p]; g[n_] := g[n] = IntegerPartitions[n];
Table[Count[g[n], p_ /; d[p] < f[p]], {n, 0, z}] (*A239954*)
Table[Count[g[n], p_ /; d[p] <= f[p]], {n, 0, z}] (*A239955*)
Table[Count[g[n], p_ /; d[p] == f[p]], {n, 0, z}] (*A239956*)
Table[Count[g[n], p_ /; d[p] > f[p]], {n, 0, z}] (*A034296*)
Table[Count[g[n], p_ /; d[p] >= f[p]], {n, 0, z}] (*A239958*)
(* second program *)
Table[Length[Select[IntegerPartitions[n], Min@@Differences[#]<-1&]], {n, 0, 30}] (* Gus Wiseman, Jun 26 2022 *)
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CROSSREFS
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Applying the condition to the conjugate gives A350839, ranked by A350841.
A116932 counts partitions with differences != -1 or -2, strict A025157.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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