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A325250
Number of integer partitions of n whose omega-sequence is strict (no repeated parts).
7
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
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.
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
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 16 2019
STATUS
approved