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A325260 Number of integer partitions of n whose omega-sequence covers an initial interval of positive integers. 4
1, 1, 2, 2, 4, 5, 5, 8, 10, 12, 13, 18, 19, 24, 25, 31, 33, 40, 40, 49, 51, 59, 60, 71, 72, 83, 84, 96, 98, 111, 111, 126, 128, 142, 143, 160, 161, 178, 179, 197, 199, 218, 218, 239, 241, 261, 262, 285, 286, 309, 310, 334, 336, 361, 361, 388, 390, 416, 417, 446 (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).

The Heinz numbers of these partitions are given by A325251.

LINKS

Table of n, a(n) for n=0..59.

FORMULA

a(n) + A325262(n) = A000041(n).

EXAMPLE

The a(1) = 1 through a(9) = 12 partitions:

  (1)  (2)   (3)   (4)    (5)    (6)    (7)     (8)     (9)

       (11)  (21)  (22)   (32)   (33)   (43)    (44)    (54)

                   (31)   (41)   (42)   (52)    (53)    (63)

                   (211)  (221)  (51)   (61)    (62)    (72)

                          (311)  (411)  (322)   (71)    (81)

                                        (331)   (332)   (441)

                                        (511)   (422)   (522)

                                        (3211)  (611)   (711)

                                                (3221)  (3321)

                                                (4211)  (4221)

                                                        (4311)

                                                        (5211)

MATHEMATICA

normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];

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

Table[Length[Select[IntegerPartitions[n], normQ[omseq[#]]&]], {n, 0, 30}]

CROSSREFS

Cf. A055932, A181819, A182850, A225486, A323014, A323023, A325250, A325251, A325262, A325277.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

Sequence in context: A277191 A035632 A114701 * A325325 A240851 A261665

Adjacent sequences:  A325256 A325258 A325259 * A325261 A325262 A325263

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 16 2019

STATUS

approved

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Last modified August 22 00:43 EDT 2019. Contains 326169 sequences. (Running on oeis4.)