cp's OEIS Frontend

A325250 Number of integer partitions of n whose omega-sequence is strict (no repeated parts).

%I A325250 #9 Aug 22 2019 09:52:54
%S A325250 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,
%T A325250 143,174,182,256,253,321,365,425,480,615,645,803,946,1180,1341,1766,
%U A325250 2021,2607,3145,3951,4727,6123,7236,9136
%N A325250 Number of integer partitions of n whose omega-sequence is strict (no repeated parts).
%C A325250 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).
%C A325250 The Heinz numbers of these partitions are given by A325247.
%F A325250 a(n) + A325262(n) = A000041(n).
%e A325250 The a(1) = 1 through a(10) = 6 partitions (A = 10):
%e A325250   1  2   3    4     5      6       7        8         9          A
%e A325250      11  111  22    11111  33      1111111  44        333        55
%e A325250               1111         222              2222      222111     3322
%e A325250                            2211             3311      111111111  4411
%e A325250                            111111           11111111             22222
%e A325250                                                                  1111111111
%t A325250 omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
%t A325250 Table[Length[Select[IntegerPartitions[n],UnsameQ@@omseq[#]&]],{n,0,30}]
%Y A325250 Cf. A047966, A181819, A323023, A325247, A325260, A325262.
%Y A325250 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).
%Y A325250 Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).
%K A325250 nonn
%O A325250 0,3
%A A325250 _Gus Wiseman_, Apr 16 2019