A238884 Number of upper triangular partitions of n.

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, 19, 20, 20, 21, 23, 23, 26, 27, 25, 26, 28, 30

Offset: 1

Author

Clark Kimberling, Mar 06 2014

Keywords

• nonn
• easy

Comments

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.

Examples

			First 12 rows of the array at 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 .. 2
52 . 10 . 4 .. 6 .. 3 .. 2
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.
		

Crossrefs

Cf. A238883, A238885, A238886, A237981, A000041.

Programs

Mathematica

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[n_] := #[[Reverse[Ordering[PadRight[Map[First[#] &, #]]]]]] &[  Tally[Map[Reverse[Sort[#]] &, Map[ut, IntegerPartitions[n]]]]]
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}]]
z = 20; Table[Flatten[u[n]], {n, 1, z}]
Flatten[Table[u[n], {n, 1, z}]]
Table[v[n], {n, 1, z}]
Flatten[Table[v[n], {n, 1, z}]] (* A238883 *)
Table[Length[v[n]], {n, 1, z}]  (* A238884 *)
(* Peter J. C. Moses, Mar 04 2014 *)