%I #13 Jan 28 2022 01:26:33
%S 1,2,2,1,3,1,1,2,2,3,4,1,4,1,1,2,3,3,5,1,1,4,1,6,3,1,2,3,1,1,3,3,3,6,
%T 2,5,1,1,5,1,4,2,6,3,1,10,2,2,2,4,1,1,2,2,2,3,5,7,2,5,1,1,7,5,6,5,1,1,
%U 4,1,6,1,6,4,2,9,3,2,3,2,4,3,10,6,1,2
%N Irregular triangular array: t(n,k) = number of times that the k-th condensed partition (in Mathematica order) of n occurs in the list of all the condensed partitions of n.
%C 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 c(p) be the partition {m(1)*x(1), m(2)*x(2), ..., x(k)*m(k)} of n. As defined at A239312, a partition q of n is a condensed partition n if q = f(p) for some partition p of n. The number of numbers in row n is A239312(n); (sum of numbers in row n) = A000041(n), and column 1 is given by A000005.
%H Clark Kimberling, <a href="/A239319/b239319.txt">Table of n, a(n) for n = 1..1000</a>
%e The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111, with respective condensations 6, 51, 42, 42, 6, 321, 33, 6, 42, 42, 6. There are A239312(6) = 5 of these, listed here in Mathematica order (x(1) >= x(2) >= ... ): 6, 51, 42, 33, 321. These occur in the list of condensed partitions with multiplicities 4, 1, 4, 1, 1, which is row 6 of the array. First 9 rows:
%e 1
%e 2
%e 2 1
%e 3 1 1
%e 2 2 3
%e 4 1 4 1 1
%e 2 3 3 5 1 1
%e 4 1 6 3 1 2 3 1 1
%e 3 3 3 6 2 5 1 1 5 1
%t z = 15; u[n_, k_] := u[n, k] = Map[Total, Split[IntegerPartitions[n][[k]]]]; r[n_] := r[n] = Table[Reverse[Sort[u[n, k]]], {k, 1, PartitionsP[n]}] ; t[n_] := t[n] = DeleteDuplicates[r[n]]; u[n_] := u[n] = Length[t[n]] ; v = Table[Count[r[n], t[n][[k]]], {n, 1, z}, {k, 1, u[n]}]; TableForm[v] (* A239319, array *)
%t Flatten[v] (* A239319, sequence *)
%Y Cf. A000041, A239312.
%K nonn,tabf,easy
%O 1,2
%A _Clark Kimberling_, Mar 15 2014