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A325334 Number of integer partitions of n with adjusted frequency depth 3 whose parts cover an initial interval of positive integers.

%I A325334 #13 Jan 17 2025 19:23:22
%S A325334 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 A325334 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 A325334 0,1,1,0,3,0,0,2,0,0,3,0,1,1,0,0,4,0,0,1,0,0,5,1,0,1,0,0,2,0,0,1,1,0,2,0,0,4
%N A325334 Number of integer partitions of n with adjusted frequency depth 3 whose parts cover an initial interval of positive integers.
%C A325334 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 A325334 The Heinz numbers of these partitions are given by A325374.
%H A325334 Antti Karttunen, <a href="/A325334/b325334.txt">Table of n, a(n) for n = 0..100000</a>
%F A325334 a(n) = A007862(n) - 1.
%e A325334 The first 30 terms count the following partitions:
%e A325334    3: (21)
%e A325334    6: (321)
%e A325334    6: (2211)
%e A325334    9: (222111)
%e A325334   10: (4321)
%e A325334   12: (332211)
%e A325334   12: (22221111)
%e A325334   15: (54321)
%e A325334   15: (2222211111)
%e A325334   18: (333222111)
%e A325334   18: (222222111111)
%e A325334   20: (44332211)
%e A325334   21: (654321)
%e A325334   21: (22222221111111)
%e A325334   24: (333322221111)
%e A325334   24: (2222222211111111)
%e A325334   27: (222222222111111111)
%e A325334   28: (7654321)
%e A325334   30: (5544332211)
%e A325334   30: (444333222111)
%e A325334   30: (333332222211111)
%e A325334   30: (22222222221111111111)
%t A325334 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A325334 unifQ[m_]:=SameQ@@Length/@Split[m];
%t A325334 Table[Length[Select[IntegerPartitions[n],normQ[#]&&!SameQ@@#&&unifQ[#]&]],{n,0,30}]
%o A325334 (PARI)
%o A325334 A007862(n) = sumdiv(n, d, ispolygonal(d, 3));
%o A325334 A325334(n) = if(!n,n,A007862(n)-1); \\ _Antti Karttunen_, Jan 17 2025
%Y A325334 Column k = 3 of A325336.
%Y A325334 Cf. A007862, A181819, A182850, A320348, A323014, A325245, A325280, A325326, A325335, A325374.
%K A325334 nonn
%O A325334 0,7
%A A325334 _Gus Wiseman_, May 01 2019
%E A325334 Data section extended to a(105) by _Antti Karttunen_, Jan 17 2025