A325282 - OEIS

%I #7 Apr 19 2019 13:42:09

%S 0,1,2,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,

%T 6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,

%U 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7

%N Maximum adjusted frequency depth among integer partitions of n.

%C The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is one 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 term "frequency depth" appears to have been coined by Clark Kimberling in A225485 and A225486, and can be applied to both integers (A323014) and integer partitions (A325280).

%C Run lengths are A325258, i.e., first differences of Levine's sequence A011784 (except at n = 1).

%F a(0) = 0; a(1) = 1; a(n > 1) = A225486(n).

%t fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];

%t Table[Max@@fdadj/@IntegerPartitions[n],{n,0,30}]

%Y Cf. A011784, A032741, A127002, A181819, A225486, A275870, A323014, A323023, A325245, A325254, A325258, A325278, A325282, A325283.

%Y Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

%K nonn

%O 0,3

%A _Gus Wiseman_, Apr 18 2019