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A325412 Number of distinct omega-sequences of integer partitions of n. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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).

LINKS

Table of n, a(n) for n=0..57.

EXAMPLE

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)

MATHEMATICA

omseq[ptn_List]:=If[ptn=={}, {}, Length/@NestWhileList[Sort[Length/@Split[#]]&, ptn, Length[#]>1&]];

Table[Length[Union[omseq/@IntegerPartitions[n]]], {n, 0, 30}]

CROSSREFS

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: A257008 A265822 A257012 * A265562 A140312 A088887

Adjacent sequences:  A325409 A325410 A325411 * A325413 A325414 A325415

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 24 2019

STATUS

approved

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Last modified October 21 11:42 EDT 2019. Contains 328296 sequences. (Running on oeis4.)