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%I #5 May 02 2019 08:53:43
%S 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 75,81,104,99,127,128,150,155,201,189,236,244,293,302,363,372,437,457,
%U 548,547,638,671,754,809,922,947,1074,1144,1290,1342,1515,1574
%N Number of integer partitions of n with adjusted frequency depth 4 whose parts cover an initial interval of positive integers.
%C 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 The Heinz numbers of these partitions are given by A325387.
%e The a(4) = 1 through a(10) = 5 partitions:
%e (211) (221) (21111) (2221) (22211) (22221) (222211)
%e (2111) (22111) (221111) (2211111) (322111)
%e (211111) (2111111) (21111111) (2221111)
%e (22111111)
%e (211111111)
%t normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
%t Table[Length[Select[IntegerPartitions[n],normQ[#]&&fdadj[#]==4&]],{n,0,30}]
%Y Column k = 4 of A325336.
%Y Cf. A000009, A007862, A181819, A182850, A317081, A320348, A323014, A325280, A325326, A325334, A325387.
%K nonn
%O 0,6
%A _Gus Wiseman_, May 01 2019