A225486 Maximal frequency depth for the partitions of n.

0, 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, 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, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Author

Clark Kimberling, May 08 2013

Keywords

• nonn

Comments

See A225485 for the definition of frequency depth.

The frequency depth of an integer partition is the number of times one must take the multiset of multiplicities to reach (1). For example, the partition (32211) has frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2) -> (1). Differs from A325282 at a(0) and a(1). - Gus Wiseman, Apr 19 2019

Examples

			(See A225485.)
		

Crossrefs

Run lengths are A325258, i.e., first differences of Levine's sequence A011784.

Cf. A008284, A116608, A181819, A182850, A182857, A225485, A323014, A323023, A325239, A325242, A325254, A325282, A325283.

Programs

Mathematica

c[s_] := c[s] = Select[Table[Count[s, i], {i, 1, Max[s]}], # > 0 &]
f[s_] := f[s] = Drop[FixedPointList[c, s], -2]
t[s_] := t[s] = Length[f[s]]
u[n_] := u[n] = Table[t[Part[IntegerPartitions[n], k]],
    {k, 1, Length[IntegerPartitions[n]]}];
Prepend[Table[Max[u[n]], {n, 2, 10}], 0]
(* second program *)
grw[q_]:=Join@@Table[ConstantArray[i,q[[Length[q]-i+1]]],{i,Length[q]}];
Join@@MapIndexed[ConstantArray[#2[[1]]-1,#1]&,Length[#]-Last[#]&/@NestList[grw,{1,1},6]] (* Gus Wiseman, Apr 19 2019 *)

Formula

a(n) = number of terms in row n of the array in A225485, for n > 0.

Extensions

More terms from Gus Wiseman, Apr 19 2019