cp's OEIS Frontend

A325335 Number of integer partitions of n with adjusted frequency depth 4 whose parts cover an initial interval of positive integers.

%I A325335 #5 May 02 2019 08:53:43
%S A325335 0,0,0,0,1,2,1,3,3,3,5,8,6,13,12,14,17,22,17,28,29,30,38,50,46,67,64,
%T A325335 75,81,104,99,127,128,150,155,201,189,236,244,293,302,363,372,437,457,
%U A325335 548,547,638,671,754,809,922,947,1074,1144,1290,1342,1515,1574
%N A325335 Number of integer partitions of n with adjusted frequency depth 4 whose parts cover an initial interval of positive integers.
%C A325335 The adjusted frequency depth of an integer partition (A325280) is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).
%C A325335 The Heinz numbers of these partitions are given by A325387.
%e A325335 The a(4) = 1 through a(10) = 5 partitions:
%e A325335   (211)  (221)   (21111)  (2221)    (22211)    (22221)     (222211)
%e A325335          (2111)           (22111)   (221111)   (2211111)   (322111)
%e A325335                           (211111)  (2111111)  (21111111)  (2221111)
%e A325335                                                            (22111111)
%e A325335                                                            (211111111)
%t A325335 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A325335 fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
%t A325335 Table[Length[Select[IntegerPartitions[n],normQ[#]&&fdadj[#]==4&]],{n,0,30}]
%Y A325335 Column k = 4 of A325336.
%Y A325335 Cf. A000009, A007862, A181819, A182850, A317081, A320348, A323014, A325280, A325326, A325334, A325387.
%K A325335 nonn
%O A325335 0,6
%A A325335 _Gus Wiseman_, May 01 2019