Number of non equivalent patterns having N clues

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Re: Number of non equivalent patterns having N clues

Postby Red Ed » Wed Jun 05, 2013 4:42 pm

PET, wow. I was aware of it but never had cause to learn & apply it. I'm impressed! 8-)
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Re: Number of non equivalent patterns having N clues

Postby Serg » Fri Jun 07, 2013 6:44 am

Hi, Red Ed!
Red Ed wrote:PET, wow. I was aware of it but never had cause to learn & apply it. I'm impressed! 8-)

Thanks. BTW, I found excellent semestral course "Combinatorial Enumeration" on the site of The University of Western Australia (see here). Lectures 5 and 7 (presentations) by Gordon Royle are devoted to Polya Enumeration Theorem.

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Re: Number of non equivalent patterns having N clues

Postby JPF » Fri Jun 07, 2013 5:38 pm

Thanks Serg for checking.

Here are # of e-d patterns having n clues in each box.

Code: Select all
   n                       Nn
                                                                 
   0                        1                                                                 
   1                      322                                                                 
   2               33,135,278                                                                 
   3           63,971,492,029                                                                 
   4        2,420,865,426,641                                                                 
   5        2,420,865,426,641                                                                 
   6           63,971,492,029                                                                 
   7               33,135,278                                                                 
   8                      322                                                                 
   9                        1     

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Re: Number of non equivalent patterns having N clues

Postby Serg » Sat Jun 08, 2013 11:02 pm

Hi, JPF!
I confirm your results. Well done!
JPF wrote:Here are # of e-d patterns having n clues in each box.

Code: Select all
   n                       Nn
                                                                 
   0                        1                                                                 
   1                      322                                                                 
   2               33,135,278                                                                 
   3           63,971,492,029                                                                 
   4        2,420,865,426,641                                                                 
   5        2,420,865,426,641                                                                 
   6           63,971,492,029                                                                 
   7               33,135,278                                                                 
   8                      322                                                                 
   9                        1     


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Re: Number of non equivalent patterns having N clues

Postby coloin » Mon Nov 10, 2014 11:17 pm

Well it seems that if you don't have an empty box there are large reductions in the possibilities
Code: Select all
1 clue in one box  - 1 pattern                             
3 clues in a band  - one clue in a box - 3 patterns   
6 clues in 2 bands - one clue in a box - 36 patterns 
8 clues in 8 boxes - 385 patterns                     
9 clues in 9 boxes - 322 patterns
C
Last edited by coloin on Tue Nov 11, 2014 10:53 pm, edited 1 time in total.
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Re: Number of non equivalent patterns having N clues

Postby Serg » Tue Nov 11, 2014 12:41 pm

Hi, coloin!
coloin wrote:Well it seems that if you don't have an empty box there are large reductions in the possibilities
Code: Select all
...
6 clues in 2 bands - one clue in a box - 32 patterns 
...

I think there are 22 essentially different patterns containing 1 clue per box for the first 6 boxes (B1-B6) and 9 clues per boxes for boxes B7-B9 (see thread One-clue-boxes patterns.

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Re: Number of non equivalent patterns having N clues

Postby coloin » Tue Nov 11, 2014 10:10 pm

Well ..... I thought I just mis-typed but I indeed missed a few
Here are 36, the first 14 have 2 empty rows in a band, followed by your 22.
Code: Select all
 1  ...............................................X..X..X....................X..X..X
 2  ...............................................X..X..X....................X..X.X.
 3  ...............................................X..X..X....................X.X..X.
 4  ...............................................X..X..X...................X..X..X.
 5  ...............................................X..X..X.................X..X..X...
 6  ...............................................X..X..X.................X..X.X....
 7  ...............................................X..X..X.................X.X..X....
 8  ...............................................X..X..X................X...X..X...
 9  ...............................................X..X..X................X...X.X....
10  ...............................................X..X..X................X..X..X....
11  ...............................................X..X..X........X.....X.....X......
12  ...............................................X..X..X........X.....X....X.......
13  ...............................................X..X..X........X....X.....X.......
14  ...............................................X..X..X.......X.....X.....X.......
15  ............................................X..X..X....................X..X..X...
16  ............................................X..X..X....................X..X.X....
17  ............................................X..X..X....................X.X..X....
18  ............................................X..X..X...................X...X..X...
19  ............................................X..X..X...................X...X.X....
20  ............................................X..X..X...................X..X..X....
21  ............................................X..X..X.................X.....X.....X
22  ............................................X..X..X.................X.....X....X.
23  ............................................X..X..X.................X....X......X
24  ............................................X..X..X.................X....X.....X.
25  ............................................X..X..X.................X.X..X.......
26  ............................................X..X..X................X.....X.....X.
27  ............................................X..X..X...........X.....X.....X......
28  ............................................X..X..X...........X.....X....X.......
29  ............................................X..X..X...........X....X.....X.......
30  ............................................X..X..X..........X......X.....X......
31  ............................................X..X..X..........X......X....X.......
32  ............................................X..X..X..........X.....X.....X.......
33  ...................................X.....X.....X..............X.....X.....X......
34  ...................................X.....X.....X..............X.....X....X.......
35  ...................................X.....X.....X..............X....X.....X.......
36  ...................................X.....X.....X.............X.....X.....X.......
C
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