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A236293 Triangular array T: T(n,1) = number of constant partitions of n; T(n,k) = number of nonconstant partitions of n that have length k, for k = 2..n-1, for n >= 2.

%I A236293 #11 Feb 06 2014 10:02:00
%S A236293 1,2,2,1,3,1,1,2,2,2,1,4,2,2,2,1,2,3,4,3,2,1,4,3,5,4,3,2,1,3,4,6,6,5,
%T A236293 3,2,1,4,4,8,9,6,5,3,2,1,2,5,10,11,10,7,5,3,2,1,6,5,11,14,13,10,7,5,3,
%U A236293 2,1,2,6,14,18,18,14,11,7,5,3,2,1,4,6,16
%N A236293 Triangular array T:  T(n,1) = number of constant partitions of n; T(n,k) = number of nonconstant partitions of n that have length k, for k = 2..n-1, for n >= 2.
%C A236293 This array occurs naturally in a method for counting the distinct cyclic permutations of the partitions of n (A236292).  The row sums, limit of column n, and limit of reversed row n are given by A000041, and (column 1) = A000005.  Does column 3 give the nonzero terms of A230059?
%H A236293 Alois P. Heinz, <a href="/A236293/b236293.txt">Rows n = 1..150, flattened</a>
%F A236293 Row n:  (d(n), f(2), f(3),..., f(n-1)), where d(n) = (number of divisors of n) = (number of constant partitions of n), and f(k) = number of nonconstant partitions of n, for k = 2,3,...,n-1.
%e A236293 First ten rows:
%e A236293 1
%e A236293 2
%e A236293 2 .. 1
%e A236293 3 .. 1 .. 1
%e A236293 2 .. 2 .. 2 .. 1
%e A236293 4 .. 2 .. 2 .. 2 .. 1
%e A236293 2 .. 3 .. 4 .. 3 .. 2 .. 1
%e A236293 4 .. 3 .. 5 .. 4 .. 3 .. 2 .. 1
%e A236293 3 .. 4 .. 6 .. 6 .. 5 .. 3 .. 2 .. 1
%e A236293 4 .. 4 .. 8 .. 9 .. 6 .. 5 .. 3 .. 2 .. 1
%e A236293 (row 6) = (4,2,2,2,1).  The 5 numbers in (4,2,2,2,1) count these partitions:  (6, 33, 222, 1111);  (51, 42); (411, 321); (3111, 2211);  (211111).  ("Constant partition" is exemplified by 6, 33, 222, 1111; i.e., all the parts are the same number.)
%t A236293 t = Map[Flatten[{Length[#[[1]]], Transpose[Tally[Map[Length, #[[2]]]]][[2]]} &[GatherBy[IntegerPartitions[#], Length[Union[#]] > 1 &]]] &, Range[3, 20]] u = Flatten[t]; Flatten[Prepend[u, {1, 2}]]
%t A236293 (* _Peter J. C. Moses_, Jan 21 2014 *)
%Y A236293 Cf. A000005, A236292, A230059.
%K A236293 nonn,tabf,easy
%O A236293 1,2
%A A236293 _Clark Kimberling_, Jan 22 2014