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

Author

Clark Kimberling, Sep 28 2023

Keywords

• nonn
• tabf

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.

Examples

			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
		

Crossrefs

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

Programs

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 *)