|
%I #8 May 02 2019 08:53:33
%S 0,0,0,1,0,0,2,0,0,1,1,0,2,0,0,2,0,0,2,0,1,2,0,0,2,0,0,1,1,0,4,0,0,1,
%T 0,0,3,0,0,1,1,0,3,0,0,3,0,0,2,0,1,1,0,0,2,1,1,1,0,0,4,0,0,2,0,0,3,0,
%U 0,1,1,0,3,0,0,2,0,0,3,0,1,1,0,0,4,0,0
%N Number of integer partitions of n with adjusted frequency depth 3 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 A325374.
%F a(n) = A007862(n) - 1.
%e The first 30 terms count the following partitions:
%e 3: (21)
%e 6: (321)
%e 6: (2211)
%e 9: (222111)
%e 10: (4321)
%e 12: (332211)
%e 12: (22221111)
%e 15: (54321)
%e 15: (2222211111)
%e 18: (333222111)
%e 18: (222222111111)
%e 20: (44332211)
%e 21: (654321)
%e 21: (22222221111111)
%e 24: (333322221111)
%e 24: (2222222211111111)
%e 27: (222222222111111111)
%e 28: (7654321)
%e 30: (5544332211)
%e 30: (444333222111)
%e 30: (333332222211111)
%e 30: (22222222221111111111)
%t normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t unifQ[m_]:=SameQ@@Length/@Split[m];
%t Table[Length[Select[IntegerPartitions[n],normQ[#]&&!SameQ@@#&&unifQ[#]&]],{n,0,30}]
%Y Column k = 3 of A325336.
%Y Cf. A007862, A181819, A182850, A320348, A323014, A325245, A325280, A325326, A325335, A325374.
%K nonn
%O 0,7
%A _Gus Wiseman_, May 01 2019
|
