OFFSET
1,3
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.
EXAMPLE
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.
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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Mar 06 2014
STATUS
approved