A237983 Triangular array read by rows: row n gives the SE partitions of n; see Comments.

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 3, 2, 1, 1, 3, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,4

Comments

See Comments at A237981 for definitions of the directional partitions, NW, NE, SW, SE. The number of SE partitions of n is A122129(n) for n >=1.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

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

Example

The first 4 rows of the array of SE partitions:
1
1 .. 1
2 .. 1 .. 1 .. 1 .. 1
3 .. 1 .. 2 .. 1 .. 1 .. 1 .. 1 .. 1 .. 1
Row 4, for example, represents the 4 NE partitions of 4 as follows:  [3,1], [2,1,1], [1,1,1,1], listed in "Mathematica order".

Mathematica

z = 10; ferrersMatrix[list_] := PadRight[Map[Table[1, {#}] &, #], {#, #} &[Max[#, Length[#]]]] &[list]; cornerPart[list_] := Module[{f = ferrersMatrix[list], u, l, ur, lr, nw, ne, se, sw}, {u, l} = {UpperTriangularize[#, 1], LowerTriangularize[#]} &[f]; {ur, lr} = {UpperTriangularize[#, 1], LowerTriangularize[#]} &[Reverse[f]]; {nw, ne, se, sw} = {Total[Transpose[u]] + Total[l], Total[ur] + Total[Transpose[lr]], Total[u] + Total[Transpose[l]], Total[Transpose[ur]] + Total[lr]};    Map[DeleteCases[Reverse[Sort[#]], 0] &, {nw, ne, se, sw}]]; cornerParts[n_] :=  Map[#[[Reverse[Ordering[PadRight[#]]]]] &, Map[DeleteDuplicates[#] &,    Transpose[Map[cornerPart, IntegerPartitions[n]]]]]; cP = Map[cornerParts, Range[z]];
Flatten[Map[cP[[#, 1]] &, Range[Length[cP]]]](*NW corner: A237981*)
Flatten[Map[cP[[#, 2]] &, Range[Length[cP]]]](*NE corner: A237982*)
Flatten[Map[cP[[#, 3]] &, Range[Length[cP]]]](*SE corner: A237983*)
Flatten[Map[cP[[#, 4]] &, Range[Length[cP]]]](*SW corner: A237982*)
(* Peter J. C. Moses, Feb 25 2014 *)

Crossrefs

Cf. A237981, A237982, A237985, A238325, A238326.

Sequence in context: A331738 A372328 A306333 * A335977 A334055 A365838

Adjacent sequences: A237980 A237981 A237982 * A237984 A237985 A237986

Keyword

nonn,tabf,easy

Author

Clark Kimberling and Peter J. C. Moses, Feb 25 2014

Status

approved