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A238884 Number of upper triangular partitions of n.

%I A238884 #7 Mar 14 2014 00:04:51
%S A238884 1,1,1,2,2,3,3,4,5,5,5,6,6,7,9,9,8,10,10,12,12,13,14,14,15,15,18,20,
%T A238884 19,20,20,21,23,23,26,27,25,26,28,30
%N A238884 Number of upper triangular partitions of n.
%C A238884 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). a(n) = number of numbers in row n of the array at A238883.
%e A238884 First 12 rows of the array at A238883:
%e A238884 1
%e A238884 2
%e A238884 3
%e A238884 4 .. 1
%e A238884 4 .. 3
%e A238884 8 .. 1 .. 2
%e A238884 10 . 3 .. 2
%e A238884 14 . 5 .. 2 .. 1
%e A238884 20 . 3 .. 4 .. 2 .. 1
%e A238884 30 . 3 .. 2 .. 1 .. 6
%e A238884 36 . 13 . 2 .. 3 .. 2
%e A238884 52 . 10 . 4 .. 6 .. 3 .. 2
%e A238884 Row 6 arises as follows:  there are 3 upper triangular (UT) partitions:  51, 33, 321, of which 51 is produced from these 8 partitions:  6, 51, 42, 411, 3111, 2211, 21111, 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.)  Since all the partitions of 6 have been used, there can be no other UT partition of 6 than 51, 33, 321.  Therefore, a(6) = 3.
%t A238884 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 A238884 t[n_] := #[[Reverse[Ordering[PadRight[Map[First[#] &, #]]]]]] &[  Tally[Map[Reverse[Sort[#]] &, Map[ut, IntegerPartitions[n]]]]]
%t A238884 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 A238884 z = 20; Table[Flatten[u[n]], {n, 1, z}]
%t A238884 Flatten[Table[u[n], {n, 1, z}]]
%t A238884 Table[v[n], {n, 1, z}]
%t A238884 Flatten[Table[v[n], {n, 1, z}]] (* A238883 *)
%t A238884 Table[Length[v[n]], {n, 1, z}]  (* A238884 *)
%t A238884 (* _Peter J. C. Moses_, Mar 04 2014 *)
%Y A238884 Cf. A238883, A238885, A238886, A237981, A000041.
%K A238884 nonn,easy
%O A238884 1,4
%A A238884 _Clark Kimberling_, Mar 06 2014