A088021 a(n) = (n^2)!/(n!)^2.

1, 1, 6, 10080, 36324288000, 1077167364120207360000, 717579719887926731226850787328000000, 23946596436219275985459662514223331478629410406400000000

Offset: 0

Author

Hugo Pfoertner, Sep 18 2003

Keywords

• nonn

Comments

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's 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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..21

Crossrefs

Cf. A001044, A088020.

Programs

Magma

[Factorial(n^2)/Factorial(n)^2: n in [0..10]]; // Vincenzo Librandi, May 31 2011

Formula

a(n) = A088020(n)/A001044(n).