|
| |
|
|
A260690
|
|
Array read by antidiagonals: D(w,h) is the maximum number of diagonals that can be placed in a w X h grid made up of unit squares when diagonals are placed in the unit squares in such a way that no two diagonals may cross or intersect at an endpoint.
|
|
0
|
|
|
|
1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 5, 5, 6, 6, 8, 8, 6, 6, 7, 7, 10, 10, 10, 7, 7, 8, 8, 12, 12, 12, 12, 8, 8, 9, 9, 14, 14, 16, 14, 14, 9, 9, 10, 10, 16, 16, 18, 18, 16, 16, 10, 10, 11, 11, 18, 18, 21, 21, 21, 18, 18, 11, 11, 12, 12, 20, 20, 24, 24, 24, 24, 20, 20, 12, 12
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
In other words, D(w,h) is the largest number of nonintersecting vertex-disjoint diagonals that can be packed in a w X h grid.
For the results for square grids (i.e., w=h), see A264041.
If at least one of the two dimensions is even, then the simple packing using the nested L-shape pattern as described in the third Pinter link at A264041 gives an optimal solution, and formulas for the number of diagonals are given below.
If, however, both dimensions are odd, it may be more difficult to find a way to pack the maximum number of diagonals or to determine what that maximum number D(w,h)is. D(w,h) is known (see Example section) for all odd-odd pairs (w,h) in which at least one dimension is less than 19.
Let L(w,h) be the number of diagonals packed in a w X h grid using the nested L-shape pattern described above, and define K(w,h) as the margin by which the number of diagonals in an optimal solution exceeds the number that would be packed using the nested L-shape pattern; i.e., K(w,h) = D(w,h) - L(w,h). If at least one of the two dimensions is even, then K(w,h) = 0; values of K(w,h) when both w and h are odd are shown in a table in the Example section.
Let B(w,h) be the number of optimal solutions (i.e., distinct configurations of D(w,h) diagonals) for a w X h grid; then known results for odd-odd pairs (w,h) include B(1,1) = 2; B(3,3) = 28; B(5,5) = 2; B(7,7) = 480, B(7,9) = 32; B(9,9) = 433284, B(9,11) = 85328, B(9,13) = 7568, B(9,15) = 256; B(11,11) = 256, ..., B(11,17) = 15813376, B(11,19) = 980224, B(11,21) = 25088. The relative scarcity of optimal solutions at a given dimension pair (w,h) may be seen as indicative of the amount of "slack" available for construction of an optimal solution at that pair; e.g., among cases with w=11, there are relatively few solutions (only 256) at h=11 (a (w,h) combination at which a K(w,h)=2 solution is just barely possible), while solutions at h values where K(w,h)=1 do not exist at all for h>21, are just possible at h=21, and become extremely plentiful as h is decreased from 21 to 13.
|
|
|
LINKS
|
|
|
|
FORMULA
|
If exactly one of the dimensions w and h is even, then D(w,h) = x*(y+1)/2 where x and y are the even and odd dimensions, respectively.
If both dimensions are even, then D(w,h) = (x/2)*(y+1) where x is the smaller dimension.
If both dimensions are odd, it can be shown (see Links) that D(w,h) >= (2*s-1)*t + floor((2*sqrt(s^2-s*t+t^2) - 2*s + t)/3) where s = (max(w,h)+1)/2 and t = (min(w,h)+1)/2.
|
|
|
EXAMPLE
|
The table begins as follows:
.
h\w| 1 2 3 4 5 6 7 8 9 10 11 12 13
---+--------------------------------------
1| 1 2 3 4 5 6 7 8 9 10 11 12 13
2| 2 3 4 5 6 7 8 9 10 11 12 13 14
3| 3 4 6 8 10 12 14 16 18 20 22 24 26
4| 4 5 8 10 12 14 16 18 20 22 24 26 28
5| 5 6 10 12 16 18 21 24 27 30 33 36 39
6| 6 7 12 14 18 21 24 27 30 33 36 39 42
7| 7 8 14 16 21 24 29 32 37 40 44 48 52
8| 8 9 16 18 24 27 32 36 40 44 48 52 56
9| 9 10 18 20 27 30 37 40 46 50 56 60 66
10|10 11 20 22 30 33 40 44 50 55 60 65 70
11|11 12 22 24 33 36 44 48 56 60 68 72 79
12|12 13 24 26 36 39 48 52 60 65 72 78 84
13|13 14 26 28 39 42 52 56 66 70 79 84 93
.
If at least one of the dimensions (w,h) is even, the exact value of D(w,h) is given by the appropriate formula in the Formula section. A table consisting of only the terms for which both dimensions are odd begins as follows:
.
h\w| 1 3 5 7 9 11 13 15 17 19 21 23 25
---+--------------------------------------------------
1| 1 3 5 7 9 11 13 15 17 19 21 23 25
3| 3 6 10 14 18 22 26 30 34 38 42 46 50
5| 5 10 16 21 27 33 39 45 51 57 63 69 75
7| 7 14 21 29 37 44 52 60 68 76 84 92 100
9| 9 18 27 37 46 56 66 76 85 95 105 115 125
11|11 22 33 44 56 68 79 91 103 115 127 138 150
13|13 26 39 52 66 79 93 107 120 134 148 162 176
15|15 30 45 60 76 91 107 122 138 154 169 185 201
17|17 34 51 68 85 103 120 138 156 173 191 209 227
19|19 38 57 76 95 115 134 154 173 193 213 232 252
21|21 42 63 84 105 127 148 169 191 213 234 256 278
23|23 46 69 92 115 138 162 185 209 232 256 280 303
25|25 50 75 100 125 150 176 201 227 252 278 303 329
.
The table below shows (with known 0 values replaced by periods and unknown values left blank, for readability) the differences by which D(w,h) exceeds the number of diagonals resulting from application of the simple nested L-shape pattern referred to above:
.
1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5
h\w| 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1
---+----------------------------------------------------
1| . . . . . . . . . . . . . . . . . . . . . . . . . .
3| . . . . . . . . . . . . . . . . . . . . . . . . . .
5| . . 1 . . . . . . . . . . . . . . . . . . . . . . .
7| . . . 1 1 . . . . . . . . . . . . . . . . . . . . .
9| . . . 1 1 1 1 1 . . . . . . . . . . . . . . . . . .
11| . . . . 1 2 1 1 1 1 1 . . . . . . . . . . . . . . .
13| . . . . 1 1 2 2 1 1 1 1 1 1 1 . . . . . . . . . . .
15| . . . . 1 1 2 2 2 2 1 1 1 1 1 1 1 1 1 . . . . . . .
17| . . . . . 1 1 2 3 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 . .
19| . . . . . 1 1 2 2 3 3 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1
21| . . . . . 1 1 1 2 3 3 3 3 2 2 2 2 2 2
23| . . . . . . 1 1 2 2 3 4 3 3 3
25| . . . . . . 1 1 2 2 3 3 4 4
27| . . . . . . 1 1 1 2 2 3 4
29| . . . . . . 1 1 1 2 2 3 5
31| . . . . . . . 1 1 2 2
33| . . . . . . . 1 1 1 2
35| . . . . . . . 1 1 1 2
37| . . . . . . . 1 1 1 2
39| . . . . . . . . 1 1
41| . . . . . . . . 1 1
43| . . . . . . . . 1 1
45| . . . . . . . . 1 1
47| . . . . . . . . 1 1
49| . . . . . . . . . 1
51| . . . . . . . . . 1
|
|
|
CROSSREFS
|
A264041 gives the terms along the main diagonal.
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
