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

Offset: 0

Author

Gus Wiseman, Apr 16 2019

Keywords

• nonn

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.

Examples

			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
		

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).

Programs

Mathematica

omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@omseq[#]&]],{n,0,30}]

Formula

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