|
| |
|
|
A237981
|
|
Array: row n gives the NW partitions of n; see Comments.
|
|
18
|
|
|
|
1, 2, 3, 4, 3, 1, 5, 4, 1, 6, 5, 1, 4, 2, 7, 6, 1, 5, 2, 8, 7, 1, 6, 2, 5, 3, 9, 8, 1, 7, 2, 6, 3, 5, 3, 1, 10, 9, 1, 8, 2, 7, 3, 6, 4, 6, 3, 1, 11, 10, 1, 9, 2, 8, 3, 7, 4, 7, 3, 1, 6, 4, 1, 12, 11, 1, 10, 2, 9, 3, 8, 4, 8, 3, 1, 7, 5, 7, 4, 1, 6, 4, 2, 13
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
Suppose that p is a partition of n, and let m = max{greatest part of p, number of parts of p}. Write the Ferrers graph of p with 1's as nodes, and pad the graph with 0's to form an m X m square matrix, which is introduced here as the Ferrers matrix of p, denoted by f(p). Four kinds of partitions are defined from f(p); they will be described by referring to the example of a 3 X 3 matrix, as follows:
...
a .. b .. c
d .. e .. f
g .. h .. i
...
Writing summands in clockwise order, the four directional partitions of p are by
NW(p) = [g + d + a + b + c, h + e + f, i]
NE(p) = [a + b + c + f + i, d + e + h, g]
SE(p) = [c + f + i + h + g, b + e + d, a]
SW(p) = [i + h + g + d + a, f + e + b, c].
The order in which the parts appear does not change the partition, but it is common to list them in nondecreasing order, as in Example 1.
...
Note that "Ferrers matrix" can be defined without reference to Ferrers graphs, as follows: an m X m matrix (x(i,j)) of 0's and 1's satisfying three properties: (1) x(1,m) = 1 or x(m,1) = 1; (2) x(i,j+1) >= x(i,j) for j=1..m-1 and i = 1..m; and (3) x(i+1,j) >= x(i,j) for i=1..m-1 and j=1..m. The number of Ferrers matrices of order m is given by A051924.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
Example 1. Let p = {6,3,3,3,1), a partition of 16. Then NW(p) = [10, 4, 2], NE(p) = [6,3,3,3,1], SE(p) = [5, 4, 3, 2, 1, 1], SW(p) = [5,4,4,1,1,1].
...
Example 2.
The first 9 rows of the array of NW partitions:
1
2
3
4 .. 3 .. 1
5 .. 4 .. 1
6 .. 5 .. 1 .. 4 .. 2
7 .. 6 .. 1 .. 5 .. 2
8 .. 7 .. 1 .. 6 .. 2 .. 5 .. 3
9 .. 8 .. 1 .. 7 .. 2 .. 6 .. 3 .. 5 .. 3 .. 1
Row 9, for example, represents the 5 NW partitions of 9 as follows: [9], [8,1], [7,2], [6,3], [5,3,1], 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*)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,tabf,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
