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A225486
Maximal frequency depth for the partitions of n.
18
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
1,2
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
FORMULA
a(n) = number of terms in row n of the array in A225485, for n > 0.
EXAMPLE
(See A225485.)
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 *)
CROSSREFS
Run lengths are A325258, i.e., first differences of Levine's sequence A011784.
Sequence in context: A136528 A263252 A276334 * A325282 A305233 A130242
KEYWORD
nonn
AUTHOR
Clark Kimberling, May 08 2013
EXTENSIONS
More terms from Gus Wiseman, Apr 19 2019
STATUS
approved