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A238885 Array:  row n gives number of times each possible lower triangular partition L(p) occurs as p ranges through the partitions of n. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 22 13:33 EDT 2017. Contains 286872 sequences.