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A325260 Number of integer partitions of n whose omega-sequence covers an initial interval of positive integers.

%I A325260 #9 Jan 13 2021 14:18:54
%S A325260 1,1,2,2,4,5,5,8,10,12,13,18,19,24,25,31,33,40,40,49,51,59,60,71,72,
%T A325260 83,84,96,98,111,111,126,128,142,143,160,161,178,179,197,199,218,218,
%U A325260 239,241,261,262,285,286,309,310,334,336,361,361,388,390,416,417,446
%N A325260 Number of integer partitions of n whose omega-sequence covers an initial interval of positive integers.
%C A325260 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 A325260 The Heinz numbers of these partitions are given by A325251.
%F A325260 a(n) + A325262(n) = A000041(n).
%F A325260 Conjectures from _Chai Wah Wu_, Jan 13 2021: (Start)
%F A325260 a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n > 9.
%F A325260 G.f.: (-x^9 - x^8 - x^7 + x^6 - x^5 - x^2 - x - 1)/((x - 1)^3*(x + 1)^2*(x^2 + 1)*(x^2 + x + 1)). (End)
%e A325260 The a(1) = 1 through a(9) = 12 partitions:
%e A325260   (1)  (2)   (3)   (4)    (5)    (6)    (7)     (8)     (9)
%e A325260        (11)  (21)  (22)   (32)   (33)   (43)    (44)    (54)
%e A325260                    (31)   (41)   (42)   (52)    (53)    (63)
%e A325260                    (211)  (221)  (51)   (61)    (62)    (72)
%e A325260                           (311)  (411)  (322)   (71)    (81)
%e A325260                                         (331)   (332)   (441)
%e A325260                                         (511)   (422)   (522)
%e A325260                                         (3211)  (611)   (711)
%e A325260                                                 (3221)  (3321)
%e A325260                                                 (4211)  (4221)
%e A325260                                                         (4311)
%e A325260                                                         (5211)
%t A325260 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A325260 omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
%t A325260 Table[Length[Select[IntegerPartitions[n],normQ[omseq[#]]&]],{n,0,30}]
%Y A325260 Cf. A055932, A181819, A182850, A225486, A323014, A323023, A325250, A325251, A325262, A325277.
%Y A325260 Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).
%K A325260 nonn
%O A325260 0,3
%A A325260 _Gus Wiseman_, Apr 16 2019