

A225485


Number of partitions of n that have frequency depth k, an array read by rows.


25



0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 3, 4, 3, 1, 1, 4, 8, 1, 1, 3, 6, 9, 3, 1, 2, 8, 12, 7, 1, 3, 11, 17, 10, 1, 1, 11, 26, 17, 1, 5, 19, 25, 27, 1, 1, 17, 44, 38, 1, 3, 25, 53, 52, 1, 1, 3, 29, 63, 76, 4
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,8


COMMENTS

Let S = {x(1),...,x(k)} be a multiset whose distinct elements are y(1),...,y(h). Let f(i) be the frequency of y(i) in S. Define F(S) = {f(1),..,f(h)}, F(1,S) = F(S), and F(m,S) = F(F(m1),S) for m>1. Then lim(F(m,S)) = {1} for every S, so that there is a least positive integer i for which F(i,S) = {1}, which we call the frequency depth of S.
Equivalently, 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).  Gus Wiseman, Apr 19 2019


LINKS

Clark Kimberling, Rows n = 1..40, concatenated


EXAMPLE

The first 9 rows:
n = 1 .... 0
n = 2 .... 1..1
n = 3 .... 1..1..1
n = 4 .... 1..2..1..1
n = 5 .... 1..1..2..3
n = 6 .... 1..3..4..3
n = 7 .... 1..1..4..8..1
n = 8 .... 1..3..6..9..3
n = 9 .... 1..2..8.12..7
For the 7 partitions of 5, successive frequencies are shown here:
5 > 1 (depth 1)
41 > 11 > 2 > 1 (depth 3)
32 > 11 > 2 > 1 (depth 3)
311 > 12 > 11 > 2 > 1 (depth 4)
221 > 12 > 11 > 2 > 1 (depth 4)
2111 > 13 > 11 > 2 > 1 (depth 4)
11111 > 5 > 1 (depth 2)
Summary: 1 partition has depth 1; 1 has depth 2; 2 have 3; and 3 have 4, so that the row for n = 5 is 1..1..2..3 .


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], i]],
{i, 1, Length[IntegerPartitions[n]]}];
Flatten[Table[Count[u[n], k], {n, 2, 25}, {k, 1, Max[u[n]]}]]


CROSSREFS

Row sums are A000041.
Column k = 2 is A032741.
Column k = 3 is A325245.
a(n!) = A325272(n).
Cf. A181819, A182850, A225486, A323014, A323023, A325238, A325239, A325254.
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (secondtolast omega), A225485 or A325280 (length/frequency depth).
Sequence in context: A325032 A270000 A029384 * A321913 A247378 A094102
Adjacent sequences: A225482 A225483 A225484 * A225486 A225487 A225488


KEYWORD

nonn,tabf


AUTHOR

Clark Kimberling, May 08 2013


STATUS

approved



