A238885 Array: row n gives number of times each possible lower triangular partition L(p) occurs as p ranges through the partitions of n.

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, 2, 2, 12, 2, 10, 2, 16, 2, 16, 8, 1, 18, 6, 4, 2, 2, 14, 2

Offset

1,2

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). A000041 = sum of numbers in row n, and A238886(n) = (number of numbers in row n) = number of lower triangular partitions of n.

Links

Clark Kimberling, Table of n, a(n) for n = 1..600

Clark Kimberling and Peter J. C. Moses,Ferrers Matrices and Related Partitions of Integers

Example

First 12 rows:
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 the 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.)

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 *)

Crossrefs

Cf. A238883, A238886, A237981, A000041.

Sequence in context: A270073 A027348 A238325 * A023566 A090970 A091972

Adjacent sequences: A238882 A238883 A238884 * A238886 A238887 A238888

Keyword

nonn,tabf,easy

Author

Clark Kimberling, Mar 06 2014

Status

approved