A136609 -id:A136609 - cp's OEIS frontend

A221976 The number of n X n matrices with zero determinant and with entries a permutation of [1,2,..,n^2].

0, 0, 2736, 8290316160

Offset: 1

R. J. Mathar, May 12 2013

This counts a subset of all (n^2)! = A088020(n) matrices which contain elements which are a permutation of [n^2]. The range of determinants is characterized in A085000, and the size of the set of different determinants in A088217.

Because any combination of row and column permutation of matrices with distinct elements generates (n!)^2 = A001044(n) different matrices, and because these restricted permutations leave the (absolute value of) the determinant constant, a(n) is a multiple of A001044(n). This factor does not yet take into account that matrix transpositions also maintain the values of determinants (and which never can be achieved by row or column permutation).

a(n) = A136609(n)*A001044(n).

A136608 (1/576)*number of ways to express n as the determinant of a 4 X 4 matrix with elements 1...16.

Original entry on oeis.org

14392910, 1550244, 2188523, 2029381, 2828486, 1905576, 2901300, 1813327, 3097897, 2169409, 2695559, 1697839, 3767494, 1682771, 2548638, 2503246, 3286048, 1684275, 3093051, 1655317, 3500693, 2374117, 2403536, 1619568

Offset: 0

Hugo Pfoertner, Jan 21 2008

0 can be expressed in a(0)*(4!)^2=8290316160 ways as the determinant of a 4 X 4 matrix which has elements 1...16. One such way is e.g. det ((1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16))=0. All numbers between -38830 and +38830 can be expressed by such a determinant. The first number not expressible is given by A088216(4). The largest expressible number is given by A085000(4)=40800.

			a(40800)=1 because the only 4X4 matrices with elements 1...16 with the determinant 40800 are the 576 combinations of determinant-preserving row and column permutations of ((16 6 4 9)(8 13 11 1)(3 12 5 14)(7 2 15 10)).

Hugo Pfoertner, Table of n, a(n) for n = 0..40800
Hugo Pfoertner, Illustration of occurrence counts. (Zoom into diagram to see details)
Hugo Pfoertner, Illustration of occurrence counts. Upper range.

Cf. A088237 [numbers not expressible by 4X4 determinant], A088215, A088216, A085000, A136609.

A309984 Number of n X n Latin squares with determinant 0, divided by 2.

Original entry on oeis.org

0, 0, 0, 16, 0, 2088, 5752, 199600889

Offset: 1

Hugo Pfoertner, Aug 26 2019

			a(4)=16: There are 2*a(4) = 32 4 X 4 Latin squares with determinant = 0, one of which is
  [1  4  3  2]
  [4  1  2  3]
  [3  2  1  4]
  [2  3  4  1].
An example of a 6 X 6 Latin square with determinant = 0 is
  [1  3  4  6  5  2]
  [3  2  6  5  4  1]
  [4  6  3  2  1  5]
  [6  5  1  3  2  4]
  [5  4  2  1  3  6]
  [2  1  5  4  6  3].

Brendan McKay, Latin squares.
Hugo Pfoertner, Occurrence counts of determinant values of Latin squares for n=1..8, zipped (2019).

Cf. A040082, A136609, A221976, A301371, A308853, A309258, A309985.

A364206 a(n) is the number of n X n nonsingular matrices using all the integers from 1 to n^2.

Original entry on oeis.org

1, 24, 360144, 20914499571840

Offset: 1

Stefano Spezia, Jul 13 2023

Right diagonal of A364203.
Cf. A085000 (maximal determinant), A350565 (minimal permanent), A350566 (maximal permanent).
Cf. A364227 (with prime numbers).
Cf. A088020, A136609, A221976.

a(n) = (n^2)! - A221976(n). - Vaclav Kotesovec, Jul 16 2023

a(4) from Vaclav Kotesovec, Jul 16 2023 (using A221976)

cp's OEIS Frontend

A221976 The number of n X n matrices with zero determinant and with entries a permutation of [1,2,..,n^2].

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A136608 (1/576)*number of ways to express n as the determinant of a 4 X 4 matrix with elements 1...16.

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A309984 Number of n X n Latin squares with determinant 0, divided by 2.

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A364206 a(n) is the number of n X n nonsingular matrices using all the integers from 1 to n^2.

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