A366063 Irregular triangle read by rows: T(n,k) is the number of partitions of n that have depth k.

1, 1, 1, 2, 1, 2, 2, 1, 3, 4, 0, 4, 6, 1, 5, 9, 1, 6, 11, 4, 1, 8, 20, 2, 0, 10, 25, 7, 0, 12, 37, 6, 1, 15, 47, 13, 2, 18, 67, 15, 1, 22, 85, 25, 3, 27, 122, 26, 1, 32, 142, 46, 10, 1, 38, 200, 53, 6, 0, 46, 259, 74, 6, 0, 54, 330, 92, 13, 1, 64, 412, 136

Offset

1,4

Comments

Suppose that P is a partition of n. Let x(1), x(2),...,x(k) be the distinct parts of P, and let m(i) be the multiplicity of x(i) in P. Let f(P) be the partition [m(1)*x(1), m(2)*x(2),...,x(k)*m(k)] of n. Define c(0,P) = P, c(1,P) = f(P), ..., c(n,P) = f(c(n-1,P), and define d(P) = least n such that c(n,P) has no repeated parts; d(P) is as the depth of P, as defined in A237685. Clearly d(P) = 0 if and only if P is a strict partition, as in A000009. Conjecture: if d >= 0, then 2^d is the least n that has a partition of depth d.

Links

Table of n, a(n) for n=1..72.

Example

First 20 rows:
   1
   1      1
   2      1
   2      2     1
   3      4     0
   4      6     1
   5      9     1
   6     11     4    1
   8     20     2    0
  10     25     7    0
  12     37     6    1
  15     47    13    2
  18     67    15    1
  22     85    25    3
  27    122    26    1
  32    142    46   10    1
  38    200    53    6    0
  46    259    74    6    0
  54    330    92   13    1
  64    412   136   15    0

Mathematica

z = 36; c[n_] := c[n] = Map[Length[FixedPointList[Sort[Map[Total, Split[#]], Greater] &, #]] - 2 &, IntegerPartitions[n]]
t = Table[Count[c[n], k], {n, 1, z}, {k, 0, Floor[Log[2, n]]}]
TableForm[t] (* this sequence as an array *)
Flatten[t]   (* this sequence *)

Crossrefs

Cf. A000009, A000041, A237685 (column 1), A237750 (column 2), A237978 (column 3), A225485 (frequency depth array).

Sequence in context: A201593 A029287 A055184 * A238190 A227925 A035388

Adjacent sequences: A366060 A366061 A366062 * A366064 A366065 A366066

Keyword

nonn,tabf

Author

Clark Kimberling, Sep 28 2023

Status

approved