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A237982 Triangular array read by rows: row n gives the NE partitions of n (see Comments). 5
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 (list; graph; refs; listen; history; text; internal format)
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 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
Sequence in context: A026792 A334301 A139100 * A239512 A334439 A036037
KEYWORD
nonn,tabf,easy
AUTHOR
STATUS
approved

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)