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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(m-1),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 (second-to-last 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

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Last modified November 18 07:22 EST 2019. Contains 329252 sequences. (Running on oeis4.)