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A325415 Number of distinct sums of omega-sequences of integer partitions of n. 4
1, 1, 2, 3, 4, 5, 8, 8, 10, 11, 13, 12, 15, 14, 16, 18, 18, 18, 21, 20, 23, 23, 24, 24, 27, 27, 28, 29, 30, 30, 34, 32, 34, 35, 36, 37, 39, 38, 40, 41, 43, 42, 45, 44, 46, 48, 48, 48, 51, 50, 53, 53, 54, 54, 57, 57, 58, 59, 60, 60, 64 (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) with sum 13.

LINKS

Table of n, a(n) for n=0..60.

EXAMPLE

The partitions of 9 organized by sum of omega sequence (first column) are:

   1: (9)

   4: (333)

   5: (81) (72) (63) (54)

   7: (621) (531) (432)

   8: (711) (522) (441)

   9: (6111) (3222) (222111)

  10: (51111) (33111) (22221) (111111111)

  11: (411111)

  12: (5211) (4311) (4221) (3321) (3111111) (2211111)

  13: (42111) (32211) (21111111)

  14: (321111)

There are a total of 11 distinct sums {1,4,5,7,8,9,10,11,12,13,14}, so a(9) = 11.

MATHEMATICA

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

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

CROSSREFS

Number of nonzero terms in row n of A325414.

Cf. A181819, A225486, A323014, A323023, A325238, A325248, A325249, A325277, A325412, A325413, A325416.

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: A297457 A297694 A297637 * A316496 A100882 A171979

Adjacent sequences:  A325412 A325413 A325414 * A325416 A325417 A325418

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 24 2019

STATUS

approved

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Last modified November 11 22:31 EST 2019. Contains 329046 sequences. (Running on oeis4.)