A325250 - OEIS

A325250

Number of integer partitions of n whose omega-sequence is strict (no repeated parts).

1, 1, 2, 2, 3, 2, 5, 2, 5, 4, 6, 2, 11, 3, 10, 12, 17, 12, 31, 22, 42, 47, 57, 60, 98, 94, 119, 143, 174, 182, 256, 253, 321, 365, 425, 480, 615, 645, 803, 946, 1180, 1341, 1766, 2021, 2607, 3145, 3951, 4727, 6123, 7236, 9136

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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 A325247.

LINKS

Table of n, a(n) for n=0..50.

FORMULA

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

EXAMPLE

The a(1) = 1 through a(10) = 6 partitions (A = 10):

1 2 3 4 5 6 7 8 9 A

11 111 22 11111 33 1111111 44 333 55

1111 222 2222 222111 3322

2211 3311 111111111 4411

111111 11111111 22222

1111111111

MATHEMATICA

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

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

CROSSREFS

Cf. A047966, A181819, A323023, A325247, A325260, A325262.

Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number).

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: A343371 A298423 A319810 * A062830 A322366 A363724

Adjacent sequences: A325247 A325248 A325249 * A325251 A325252 A325253

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 16 2019

STATUS

approved