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Subject: Number of four by four permutations... Author: Jed Pack [ Next Thread | Previous Thread | Next Message | Previous Message ] Date Posted: 18:35:25 04/17/02 Wed Steven Cullinane, Your website states: In the 4x4 case, D is a four-diamond figure (left, below) and G is a group of 322,560 permutations generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2x2 quadrants. Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry. I wonder how you got the number 322,560. I suggest the number is at most 30^2=900. Please let me know if you find the error in my logic. In the original four diamond configuration. The matrix indicating which squares have the black corner either at top-right or top-left is: 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 Similarly, the matrix indicating which squares have the black corner either at top-right or bottom-right is: 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 These two matricies fully describe the configuration. Swapping columns, rows, and quadrants can change these matricies into any of the following other 30 matricies (there are no other possibilities): 0 0 1 1 | 0 1 1 0 | 0 0 1 1 | 1 1 1 1 | 1 0 1 0 | 0 0 1 1 | 0 1 1 0 | 1 1 0 0 | 0 0 0 0 | 1 0 1 0 | 0 0 1 1 | 1 0 0 1 | 1 1 0 0 | 0 0 0 0 | 0 1 0 1 | 0 0 1 1 | 1 0 0 1 | 0 0 1 1 | 1 1 1 1 | 0 1 0 1 | 0 1 0 1 | 0 0 0 0 | 0 1 0 1 | 1 0 0 1 | 1 1 0 0 | 0 1 0 1 | 1 1 1 1 | 1 0 1 0 | 0 1 1 0 | 1 1 0 0 | 0 1 0 1 | 0 0 0 0 | 1 0 1 0 | 1 0 0 1 | 0 0 1 1 | 0 1 0 1 | 1 1 1 1 | 0 1 0 1 | 0 1 1 0 | 0 0 1 1 | 0 1 1 0 | 0 0 1 1 | 0 1 1 0 | 1 0 1 0 | 1 1 1 1 | 0 1 1 0 | 1 1 0 0 | 1 0 0 1 | 0 1 0 1 | 1 1 1 1 | 0 1 1 0 | 0 0 1 1 | 1 0 0 1 | 1 0 1 0 | 0 0 0 0 | 0 1 1 0 | 1 1 0 0 | 0 1 1 0 | 0 1 0 1 | 0 0 0 0 | 0 0 0 0 | 0 1 0 1 | 1 0 0 1 | 1 1 0 0 | 1 0 0 1 | 0 0 0 0 | 1 0 1 0 | 0 1 1 0 | 0 0 1 1 | 1 0 0 1 | 1 1 1 1 | 0 1 0 1 | 0 1 1 0 | 1 1 0 0 | 1 0 0 1 | 1 1 1 1 | 1 0 1 0 | 1 0 0 1 | 0 0 1 1 | 1 0 0 1 | 0 0 1 1 | 0 1 1 0 | 1 0 1 0 | 1 1 1 1 | 1 0 1 0 | 0 0 1 1 | 1 0 0 1 | 0 1 0 1 | 0 0 0 0 | 1 0 1 0 | 1 1 0 0 | 0 1 1 0 | 0 1 0 1 | 1 1 1 1 | 1 0 1 0 | 1 1 0 0 | 1 0 0 1 | 1 0 1 0 | 0 0 0 0 | 1 0 1 0 | 0 1 0 1 | 0 0 0 0 | 1 1 0 0 | 1 0 0 1 | 1 1 0 0 | 0 1 0 1 | 1 1 1 1 | 0 0 1 1 | 1 0 0 1 | 1 1 0 0 | 1 0 1 0 | 1 1 1 1 | 0 0 1 1 | 0 1 1 0 | 1 1 0 0 | 1 0 1 0 | 0 0 0 0 | 1 1 0 0 | 0 1 1 0 | 1 1 0 0 | Consequently, any pattern that can be obtained through such transformations can be described by a pair of these two matricies. There are 30^2 such pairs, and hence can be no more than 30^2 patterns obtained through the mentioned transformations. [ Next Thread | Previous Thread | Next Message | Previous Message ]

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[> Subject: Re: Number of four by four permutations... Author: S. H. Cullinane [ Edit | View ] Date Posted: 00:16:40 04/25/02 Thu Dear Mr. Pack: Here's how I got the number 322,560: "G is isomorphic to the affine group A on the linear 4-space over GF(2)," as stated in my website. I suggest you consult a book on finite geometry and group theory such as Geometry and Symmetry, by Paul Yale, to learn more about what the above sentence means. It is a nontrivial exercise to PROVE the sentence. Your error seems to be in your statement that "these two matrices fully describe the configuration." Try describing the configuration with two orthogonal 4x4 Latin squares instead. Yours truly, S. H. Cullinane [ Post a Reply to This Message ]