A238886 Number of lower triangular partitions of n.

1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 38, 40, 43, 45, 48, 50, 53, 55, 58, 60, 63, 66, 69, 72, 75, 78

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 diagonal, respectively, of the Ferrers matrix of p defined at A237981. The lower triangular partition of p, denoted by L(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 A238885.

Examples

			First 12 rows of A238885:
1
2
2 .. 1
2 .. 3
2 .. 2 .. 3
2 .. 2 .. 6 .. 1
2 .. 2 .. 6 .. 1 .. 4
2 .. 2 .. 8 .. 2 .. 4 .. 4
2 .. 2 .. 8 .. 2 .. 6 .. 1 .. 8 .. 1
2 .. 2 .. 10 . 2 .. 6 .. 2 .. 12 . 4 .. 2
2 .. 2 .. 10 . 2 .. 8 .. 2 .. 12 . 1 .. 12 . 4 .. 1
2 .. 2 .. 12 . 2 .. 8 .. 2 .. 16 . 2 .. 12 . 6 .. 9 .. 4
Row 4 arises as follows:  there are 3 lower triangular (LT) partitions:  41, 311, 221, of which 41 is produced from these 2 partitions: 5 and 11111; while the LT partition 311 is produced by 41 and 2111, and the LT partition 221 is produced by 32, 311, 221; thus row 5 is 2, 2, 3.  (For example, the rows of the Ferrers matrix of 311 are (1,1,1), (1,0,0), (1,0,0), with principal diagonal (1,0,0), so that u = 2, v = 1, w = 2; as a partition, 212 is identical to 221.)  Since all the partitions of 5 have been used, there can be no other LT partition of 5 than 41, 311, 221.  Therefore, a(5) = 3.
		

Crossrefs

Cf. A238883, A238885, A237981, A000041.

Programs

Mathematica

ferrersMatrix[list_] := PadRight[Map[Table[1, {#}] &, #], {#, #} &[Max[#, Length[#]]]] &[list]; lt[list_] := Select[Map[Total[Flatten[#]] &, {LowerTriangularize[#, -1], Diagonal[#], UpperTriangularize[#, 1]}] &[ferrersMatrix[list]], # > 0 &]; t[n_] := #[[Reverse[Ordering[PadRight[Map[First[#] &, #]]]]]] &[Tally[Map[Reverse[Sort[#]] &, Map[lt, 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 = 10; 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}]]  (* A238885 *)
Table[Length[v[n]], {n, 1, z}]  (* A238886 *)
(* Peter J. C. Moses, Mar 04 2014 *)