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A237983 Triangular array read by rows: row n gives the SE partitions of n; see Comments. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A071974 A056622 A306333 * A254613 A129265 A030358

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

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Last modified February 22 16:10 EST 2019. Contains 320399 sequences. (Running on oeis4.)