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
Examples
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".
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
Programs
-
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 *)
Comments