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

%I A366063 #13 Feb 25 2025 20:43:54
%S A366063 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,
%T A366063 6,1,15,47,13,2,18,67,15,1,22,85,25,3,27,122,26,1,32,142,46,10,1,38,
%U A366063 200,53,6,0,46,259,74,6,0,54,330,92,13,1,64,412,136
%N A366063 Irregular triangle read by rows: T(n,k) is the number of partitions of n that have depth k.
%C A366063 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.
%e A366063 First 20 rows:
%e A366063    1
%e A366063    1      1
%e A366063    2      1
%e A366063    2      2     1
%e A366063    3      4     0
%e A366063    4      6     1
%e A366063    5      9     1
%e A366063    6     11     4    1
%e A366063    8     20     2    0
%e A366063   10     25     7    0
%e A366063   12     37     6    1
%e A366063   15     47    13    2
%e A366063   18     67    15    1
%e A366063   22     85    25    3
%e A366063   27    122    26    1
%e A366063   32    142    46   10    1
%e A366063   38    200    53    6    0
%e A366063   46    259    74    6    0
%e A366063   54    330    92   13    1
%e A366063   64    412   136   15    0
%t A366063 z = 36; c[n_] := c[n] = Map[Length[FixedPointList[Sort[Map[Total, Split[#]], Greater] &, #]] - 2 &, IntegerPartitions[n]]
%t A366063 t = Table[Count[c[n], k], {n, 1, z}, {k, 0, Floor[Log[2, n]]}]
%t A366063 TableForm[t] (* this sequence as an array *)
%t A366063 Flatten[t]   (* this sequence *)
%Y A366063 Cf. A000009, A000041, A237685 (column 1), A237750 (column 2), A237978 (column 3), A225485 (frequency depth array).
%K A366063 nonn,tabf
%O A366063 1,4
%A A366063 _Clark Kimberling_, Sep 28 2023