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A325412
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Number of distinct omega-sequences of integer partitions of n.
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3
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1, 1, 2, 3, 5, 5, 10, 9, 14, 15, 20, 21, 33, 30, 39, 45, 54, 54, 69, 68, 85, 90, 100, 104, 128, 127, 141, 153, 172, 175, 205, 203, 229, 240, 257, 274, 308, 309, 335, 356, 390, 395, 437, 444, 481, 506, 530, 549, 602, 609, 648, 672, 710, 727, 777, 798, 848, 871
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OFFSET
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0,3
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COMMENTS
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The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1).
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LINKS
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Table of n, a(n) for n=0..57.
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EXAMPLE
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The a(1) = 1 through a(9) = 15 omega-sequences:
(1) (1) (1) (1) (1) (1) (1) (1) (1)
(21) (31) (21) (51) (21) (71) (21) (31)
(221) (41) (221) (31) (221) (41) (91)
(221) (3221) (61) (331) (81) (221)
(3221) (4221) (221) (3221) (221) (331)
(331) (4221) (331) (621)
(421) (5221) (421) (3221)
(3221) (6221) (3221) (4221)
(4221) (43221) (4221) (5221)
(5221) (5221) (6221)
(6221) (7221)
(7221) (8221)
(43221) (43221)
(53221) (53221)
(63221)
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MATHEMATICA
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omseq[ptn_List]:=If[ptn=={}, {}, Length/@NestWhileList[Sort[Length/@Split[#]]&, ptn, Length[#]>1&]];
Table[Length[Union[omseq/@IntegerPartitions[n]]], {n, 0, 30}]
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CROSSREFS
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Cf. A181819, A225486, A323014, A323023, A325238, A325248, A325277, A325413, A325415.
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (frequency depth), A325414 (omega-sequence sum).
Sequence in context: A338750 A265822 A257012 * A265562 A140312 A088887
Adjacent sequences: A325409 A325410 A325411 * A325413 A325414 A325415
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KEYWORD
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nonn
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AUTHOR
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Gus Wiseman, Apr 24 2019
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STATUS
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approved
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