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A225486 Maximal frequency depth for the partitions of n.

%I A225486 #12 Apr 22 2019 14:26:18
%S A225486 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,
%T A225486 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,
%U A225486 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7
%N A225486 Maximal frequency depth for the partitions of n.
%C A225486 See A225485 for the definition of frequency depth.
%C A225486 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
%F A225486 a(n) = number of terms in row n of the array in A225485, for n > 0.
%e A225486 (See A225485.)
%t A225486 c[s_] := c[s] = Select[Table[Count[s, i], {i, 1, Max[s]}], # > 0 &]
%t A225486 f[s_] := f[s] = Drop[FixedPointList[c, s], -2]
%t A225486 t[s_] := t[s] = Length[f[s]]
%t A225486 u[n_] := u[n] = Table[t[Part[IntegerPartitions[n], k]],
%t A225486     {k, 1, Length[IntegerPartitions[n]]}];
%t A225486 Prepend[Table[Max[u[n]], {n, 2, 10}], 0]
%t A225486 (* second program *)
%t A225486 grw[q_]:=Join@@Table[ConstantArray[i,q[[Length[q]-i+1]]],{i,Length[q]}];
%t A225486 Join@@MapIndexed[ConstantArray[#2[[1]]-1,#1]&,Length[#]-Last[#]&/@NestList[grw,{1,1},6]] (* _Gus Wiseman_, Apr 19 2019 *)
%Y A225486 Run lengths are A325258, i.e., first differences of Levine's sequence A011784.
%Y A225486 Cf. A008284, A116608, A181819, A182850, A182857, A225485, A323014, A323023, A325239, A325242, A325254, A325282, A325283.
%K A225486 nonn
%O A225486 1,2
%A A225486 _Clark Kimberling_, May 08 2013
%E A225486 More terms from _Gus Wiseman_, Apr 19 2019