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.
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
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Sep 28 2023
STATUS
approved