A237982 Triangular array read by rows: row n gives the NE partitions of n (see Comments).

1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 4, 3, 1, 2, 1, 1, 1, 1, 1, 1, 5, 4, 1, 3, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 1, 4, 2, 4, 1, 1, 3, 2, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 6, 1, 5, 2, 5, 1, 1, 4, 3, 4, 2, 1, 4, 1, 1

Offset

1,2

Comments

See Comments at A237981 for definitions of the directional partitions, NW, NE, SW, SE. The number of NE partitions of n, and also the number of SW partitions of n, is A237329(n), for n >=1.

The order is: each partition has nonincreasing parts and the partitions are ordered anti-lexicographic (called "Mathematica order" in the example). - Wolfdieter Lang, Mar 21 2014

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 NW partitions:
1
2 .. 1 .. 1
3 .. 2 .. 1 .. 1 .. 1 .. 1
4 .. 3 .. 1 .. 2 .. 1 .. 1 .. 1 .. 1 .. 1 .. 1
Row 4, for example, represents the 4 NE partitions of 4 as follows:  [4], [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, A237329, A237983, A237985, A238325, A238326.

Sequence in context: A026792 A334301 A139100 * A239512 A334439 A036037

Adjacent sequences: A237979 A237980 A237981 * A237983 A237984 A237985

Keyword

nonn,tabf,easy

Author

Clark Kimberling and Peter J. C. Moses, Feb 23 2014

Status

approved