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A238883 Array: row n gives number of times each upper triangular partition U(p) occurs as p ranges through the partitions of n.

%I A238883 #11 Mar 14 2014 00:03:56
%S A238883 1,2,3,4,1,4,3,8,1,2,10,3,2,14,5,2,1,20,3,4,2,1,30,3,2,1,6,36,13,2,3,
%T A238883 2,52,10,4,6,3,2,70,9,9,4,6,3,94,16,6,5,10,2,2,122,24,4,8,1,12,2,2,1,
%U A238883 160,33,4,12,6,4,9,2,1,206,37,18,14,6,2,6,8
%N A238883 Array:  row n gives number of times each upper triangular partition U(p) occurs as p ranges through the partitions of n.
%C A238883 Suppose that p is a partition.  Let u, v, w be the number of 1's above, on, and below the principal antidiagonal, respectively, of the Ferrers matrix of p defined at A237981.  The upper triangular partition of p, denoted by U(p), is {u,v} if w = 0 and {u,v,w} otherwise.  In row n, the counted partitions are taken in Mathematica order (i.e., reverse lexicographic).  A000041 = sum of numbers in row n, and A238884(n) = (number of numbers in row n) = number of upper triangular partitions of n.
%H A238883 Clark Kimberling, <a href="/A238883/b238883.txt">Table of n, a(n) for n = 1..300</a>
%H A238883 Clark Kimberling and Peter J. C. Moses, <a href="http://faculty.evansville.edu/ck6/GalleryThree/Introduction3.html">Ferrers Matrices and Related Partitions of Integers</a>
%e A238883 First 12 rows:
%e A238883 1
%e A238883 2
%e A238883 3
%e A238883 4 .. 1
%e A238883 4 .. 3
%e A238883 8 .. 1 .. 2
%e A238883 10 . 3 .. 2
%e A238883 14 . 5 .. 2 .. 1
%e A238883 20 . 3 .. 4 .. 2 .. 1
%e A238883 30 . 3 .. 2 .. 1 .. 6
%e A238883 36 . 13 . 2 .. 3 .. 2
%e A238883 52 . 10 . 4 .. 6 .. 3 .. 2
%e A238883 Row 6 arises as follows:  there are 3 upper triangular (UT) partitions:  51, 33, 321, of which 51 is produced from the 8 partitions  6, 51, 42, 411, 3111, 2211, 21111, and 111111, while the UT partition 33 is produced from the single partition 321, and the only other UT partition of 6, namely 321, is produced from the partitions 33 and 222.  (For example, the rows of the Ferrers matrix of 222 are (1,1,0), (1,1,0), (1,1,0), with principal antidiagonal (0,1,1), so that u = 3, v = 2, w = 1.)
%t A238883 ferrersMatrix[list_] := PadRight[Map[Table[1, {#}] &, #], {#, #} &[Max[#, Length[#]]]] &[list]; ut[list_] := Select[Map[Total[Flatten[#]] &, {LowerTriangularize[#, -1], Diagonal[#], UpperTriangularize[#, 1]}] &[Reverse[ferrersMatrix[list]]], # > 0 &];
%t A238883 t[n_] := #[[Reverse[Ordering[PadRight[Map[First[#] &, #]]]]]] &[  Tally[Map[Reverse[Sort[#]] &, Map[ut, IntegerPartitions[n]]]]]
%t A238883 u[n_] := Table[t[n][[k]][[1]], {k, 1, Length[t[n]]}]; v[n_] := Table[t[n][[k]][[2]], {k, 1, Length[t[n]]}]; TableForm[Table[t[n], {n, 1, 12}]]
%t A238883 z = 20; Table[Flatten[u[n]], {n, 1, z}]
%t A238883 Flatten[Table[u[n], {n, 1, z}]]
%t A238883 Table[v[n], {n, 1, z}]
%t A238883 Flatten[Table[v[n], {n, 1, z}]] (* A238883 *)
%t A238883 Table[Length[v[n]], {n, 1, z}]  (* A238884 *)
%t A238883 (* _Peter J. C. Moses_, Mar 04 2014 *)
%Y A238883 Cf. A238884, A238885, A238886, A237981, A000041.
%K A238883 nonn,tabf,easy
%O A238883 1,2
%A A238883 _Clark Kimberling_, Mar 06 2014