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A088021
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a(n) = (n^2)!/(n!)^2.
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9
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OFFSET
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0,3
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COMMENTS
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Based on an observation of Hugo Pfoertner, W. Edwin Clark, conjectured and Xiang-dong Hou proved that (n^2)!/(n!)^2 gives the number of distinct determinants of the generic n X n matrix whose entries are n^2 different indeterminates under all (n^2)! permutations of the entries.
Using J. T. Schwarz' Sparse Zeros Lemma this implies that for any positive integer n there is an n X n matrix A with positive integer entries such that the set of determinant values obtained from A by permuting the elements of A is (n^2)!/(n!)^2.
Moreover, for any entries, no larger number of determinants can be obtained. In fact, by the Sparse Zeros Lemma one can select the entries of A from any sufficiently large subset of real numbers.
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LINKS
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FORMULA
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PROG
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(Magma) [Factorial(n^2)/Factorial(n)^2: n in [0..10]]; // Vincenzo Librandi, May 31 2011
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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