A322576 -id:A322576 - cp's OEIS frontend

A088216 Smallest nonnegative number not expressible as determinant of an n X n matrix with elements 1..n^2.

0, 324, 38831, 6773999, 1859163031

Offset: 2

Hugo Pfoertner, Sep 23 2003

			a(2)=0 because the 2 X 2 determinant of a matrix with entries that are permutations of 1,2,3,4 can only assume the values +-2,+-5,+-10.

Hugo Pfoertner, Find all possible values of det(A) for matrix with elements 1..n^2. FORTRAN program.

a(3)=A088214(1), a(4)=A088237(1), a(5)=A088238(1), a(6)=A325900(1).

Cf. A085000, A088215, A088217, A322576.

a(6) from Hugo Pfoertner, Sep 07 2019

A325900 Numbers less than the maximum possible determinant A085000(6)=1865999570 not occurring as determinant of a 6 X 6 matrix with entries {1,..,36}.

Original entry on oeis.org

1859163031, 1859166733, 1859193211, 1859235497, 1859254067, 1859268659, 1859282869, 1859288597, 1859291519, 1859294309, 1859309245, 1859317037, 1859320819, 1859324083, 1859324501, 1859331797, 1859333683, 1859335879, 1859348273, 1859348639, 1859351059, 1859358869

Offset: 1

Hugo Pfoertner, Sep 07 2019

There are 4521437 terms in the sequence.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A085000, A088214, A088216, A088237, A088238, A322576.

A327281 Numbers less than the maximum possible determinant A301371(8)=27296640 not occurring as determinant of an 8 X 8 matrix whose entries are a permutation of the multiset {1^8,..,8^8}.

Original entry on oeis.org

27003797, 27011623, 27012187, 27012757, 27012835

Offset: 1

Hugo Pfoertner, Sep 20 2019

The sequence terms are based on numerical results. No proof for the non-existence of a matrix with given determinant value less than Gasper's upper bound (see Corollary 3 in Sigg) is known. The number of sequence terms is <= 205426. Candidates for a continuation of the sequence are provided as external file.

			The following matrices have determinants in the vicinity of a(1) = A322576(8) = 27003797, for which no corresponding matrix is known:
27003795 = det[2,5,1,4,8,7,3,6; 3,2,4,8,5,5,8,2; 7,1,7,3,4,8,2,4; 8,7,1,4,2,5,6,3; 1,6,7,3,1,6,6,6; 4,8,7,5,6,3,2,1; 5,3,5,1,7,1,7,6; 5,4,4,8,3,2,2,8],
27003796 = det[1,5,6,3,7,4,8,2; 6,4,8,2,2,8,3,3; 4,1,2,3,6,7,5,8; 5,5,2,8,6,7,3,1; 8,6,2,3,2,3,8,4; 1,6,5,7,1,4,4,7; 5,8,4,2,7,3,1,6; 6,1,7,7,5,1,4,5],
27003798 = det[7,4,2,8,7,3,1,5; 3,6,6,1,8,2,4,6; 2,1,3,5,6,8,6,5; 6,5,7,3,3,8,1,3; 5,2,8,6,4,2,7,2; 8,3,3,2,2,4,6,8; 1,7,5,7,1,4,4,7; 5,8,1,4,4,5,7,1],
27003799 = det[2,8,6,4,7,1,5,4; 5,7,1,8,1,4,6,4; 5,3,7,3,3,6,8,1; 2,3,8,6,2,5,2,7; 3,4,2,1,5,7,6,8; 8,7,5,2,4,6,1,4; 3,3,3,7,8,7,2,2; 8,1,4,5,6,1,5,6].

Table of n, a(n) for n = 1..5
Hugo Pfoertner, Plot of sequence.
Hugo Pfoertner, Conjectured continuation of A327281, (2019).
Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 2018.

Cf. A301371, A322576, A325900.

A309799 Number of distinct nonnegative values that can be assumed by the determinant of an n X n matrix whose entries are a permutation of the multiset {1^n,..,n^n}.

Original entry on oeis.org

1, 2, 13, 147, 2162, 40498, 948618

Offset: 1

Hugo Pfoertner, Aug 29 2019

a(8) >= 27091220. - Hugo Pfoertner, Sep 23 2019

			a(2) = 2: 0 = det[1,1; 2,2], 3 = det[2,1; 1,2] are the two possible nonnegative values of the determinant.
a(3) = 13, because
   0 = det[1,2,3; 1,2,3; 1,2,3],  1 = det[2,2,1; 3,2,1; 3,3,1],
   2 = det[3,2,3; 1,2,3; 1,1,2],  3 = det[3,3,3; 1,2,2; 1,1,2],
   4 = det[1,3,3; 2,2,1; 1,3,2],  5 = det[2,2,1; 1,3,3; 1,2,3],
   6 = det[1,3,2; 1,2,3; 2,1,3],  7 = det[1,3,1; 1,2,3; 2,2,3],
   8 = det[1,1,2; 3,3,2; 1,3,2], 12 = det[2,3,1; 2,1,3; 3,1,2],
  13 = det[3,3,1; 1,3,2; 2,1,2], 15 = det[2,1,3; 3,1,1; 2,3,2],
  18 = det[2,3,1; 1,2,3; 3,1,2]
are the 13 possible nonnegative values of the determinant.

Cf. A088216, A088217, A301371, A309258, A322576, A327281.

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A088216 Smallest nonnegative number not expressible as determinant of an n X n matrix with elements 1..n^2.

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A325900 Numbers less than the maximum possible determinant A085000(6)=1865999570 not occurring as determinant of a 6 X 6 matrix with entries {1,..,36}.

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A327281 Numbers less than the maximum possible determinant A301371(8)=27296640 not occurring as determinant of an 8 X 8 matrix whose entries are a permutation of the multiset {1^8,..,8^8}.

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A309799 Number of distinct nonnegative values that can be assumed by the determinant of an n X n matrix whose entries are a permutation of the multiset {1^n,..,n^n}.

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