A085000 -id:A085000 - cp's OEIS frontend

A088214 Numbers less than the maximum possible determinant A085000(3)=412 not occurring as determinant of a 3 X 3 matrix with elements 1..9.

324, 329, 355, 357, 358, 362, 364, 365, 367, 373, 375, 378, 381, 383, 386, 387, 394, 397, 399, 401, 403, 406, 409, 411

Offset: 1

Hugo Pfoertner, Sep 23 2003

			398 is not in the sequence because it can be expressed as det ((9 3 5)(4 8 2)(1 6 7)).

Hugo Pfoertner, Possible determinants of 3 X 3 matrix with elements 1..9.

Cf. A085000, A088215, A088216, A088217.

A088237 Numbers less than the maximum possible determinant A085000(4)=40800 not occurring as determinant of a 4 X 4 matrix with elements 1..16.

Original entry on oeis.org

38831, 38875, 38959, 38963, 39013, 39057, 39059, 39061, 39063, 39071, 39099, 39109, 39111, 39125, 39137, 39154, 39155, 39178, 39190, 39191, 39205, 39223, 39245, 39247, 39251, 39254, 39267, 39274, 39277, 39279, 39281, 39297, 39310

Offset: 1

Hugo Pfoertner, Sep 25 2003

Hugo Pfoertner, Table of n, a(n) for n = 1..1073 [gives all terms]

Cf. A085000, A088214, A088216.

Full sequence from Hugo Pfoertner, Aug 31 2014

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.

A088238 Numbers less than the maximum possible determinant A085000(5)=6839492 not occurring as determinant of a 5X5 matrix with elements 1..25.

Original entry on oeis.org

6773999, 6774223, 6774529, 6775471, 6775491, 6775877, 6776023, 6776291, 6776373, 6776557, 6776779, 6776803, 6777487, 6777655, 6777718, 6777731, 6778001, 6778103, 6778111, 6778781, 6778909, 6779065, 6779123, 6779261

Offset: 1

Hugo Pfoertner, Oct 03 2003

The first term of this sequence is A088216(5).

The sequence contains exactly 38298 terms.

Hugo Pfoertner, Table of n, a(n) for n = 1..38298 [Gives all terms]
Hugo Pfoertner, Find all possible values of det(A) for matrix with elements 1..n^2. FORTRAN program.

Cf. A085000, A088214, A088237, A088216, A088217.

Fortran
```
c See link.
```

Full sequence from Hugo Pfoertner, Aug 31 2014

A301371 Maximum determinant of an n X n matrix with n copies of the numbers 1 .. n.

Original entry on oeis.org

1, 1, 3, 18, 160, 2325, 41895, 961772, 27296640, 933251220

Offset: 0

Hugo Pfoertner, Mar 21 2018

929587995 <= a(9) <= 934173632 (upper bound from Gasper's determinant theorem). The lower bound corresponds to a Latin square provided in A309985, but it is unknown whether a larger determinant value can be achieved by an unconstrained arrangement of the matrix entries. - Hugo Pfoertner, Aug 27 2019

Oleg Vlasii found a 9 X 9 matrix significantly exceeding the determinant value achievable by a Latin square. See example and links. - Hugo Pfoertner, Nov 04 2020

			Matrices with maximum determinants:
a(2) = 3:
  (2  1)
  (1  2)
a(3) = 18:
  (3  1  2)
  (2  3  1)
  (1  2  3)
a(4) = 160:
  (4  3  2  1)
  (1  4  3  2)
  (3  1  4  3)
  (2  2  1  4)
a(5) = 2325:
  (5  3  1  2  4)
  (2  5  4  1  3)
  (4  1  5  3  2)
  (3  4  2  5  1)
  (1  2  3  4  5)
a(6) = 41895:
  (6  1  4  2  3  5)
  (3  6  2  1  5  4)
  (4  5  6  3  2  1)
  (5  3  1  6  4  2)
  (1  2  5  4  6  3)
  (2  4  3  5  1  6)
a(7) = 961772:
  (7  2  3  5  1  4  6)
  (3  7  6  4  2  1  5)
  (2  1  7  6  4  5  3)
  (4  5  1  7  6  3  2)
  (6  3  5  1  7  2  4)
  (5  6  4  2  3  7  1)
  (1  4  2  3  5  6  7)
a(8) = 27296640:
  (8  8  3  5  4  3  4  1)
  (1  8  6  3  1  6  6  5)
  (5  3  8  1  7  6  4  2)
  (5  1  6  8  2  4  7  3)
  (1  5  2  7  8  6  4  3)
  (7  3  2  4  3  8  2  7)
  (5  4  2  2  6  2  8  7)
  (4  5  7  6  5  1  1  7)
a(n) is an upper bound for the determinant of an n X n Latin square. a(n) = A309985(n) for n <= 7. a(8) > A309985(8). - _Hugo Pfoertner_, Aug 26 2019
From _Hugo Pfoertner_, Nov 04 2020: (Start)
a(9) = 933251220, achieved by a Non-Latin square:
  (9  5  5  3  3  2  2  8  8)
  (4  9  2  6  7  5  3  1  8)
  (4  7  9  2  1  8  6  3  5)
  (6  3  7  9  4  1  8  2  5)
  (6  2  8  5  9  7  1  4  3)
  (7  4  1  8  2  9  5  6  3)
  (7  6  3  1  8  4  9  5  2)
  (1  8  6  7  5  3  4  9  2)
  (1  1  4  4  6  6  7  7  9)
found by Oleg Vlasii as an answer to the IBM Ponder This Challenge November 2019. See links. (End)

Ortwin Gasper, Hugo Pfoertner and Markus Sigg, An Upper Bound for the Determinant of a Matrix with given Entry Sum and Square Sum, JIPAM, Journal of Inequalities in Pure and Applied Mathematics, Volume 10, Issue 3, Article 63, 2008.
IBM Research, Large 9x9 determinant, Ponder This Challenge November 2019.
Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 2018.
Oleg Vlasii, Determinant-OEIS-A301371-9, program and description, 4 Dec 2019.
Index entries for sequences related to maximal determinants

Cf. A085000, A309985, A328030, A328031.

A328030(n) <= a(n) <= A328031(n). - Hugo Pfoertner, Nov 04 2019

a(8) from Hugo Pfoertner, Aug 26 2019

a(9) from Hugo Pfoertner, Nov 04 2020

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

Original entry on oeis.org

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

A088217 Number of distinct values that can be assumed by the determinant of an n X n matrix whose entries are all permutations of the numbers 1..n^2.

Original entry on oeis.org

1, 6, 777, 79455, 13602389, 3722956267

Offset: 1

Hugo Pfoertner, Sep 23 2003

a(5) = 1 + 2*(A085000(5) - (number of terms of A088238)).

			a(2)=6 because the determinants of the 24 2 X 2 matrices whose entries are all permutations of 1,2,3,4 can only assume the values -10,-5,-2,2,5,10.

Cf. A085000, A088214, A088215, A088216, A088238, A325900.

Fortran
```
C See link given in A088238.
```

f[n_] := (p = Permutations[ Table[i, {i, n^2}]]; Length[ Union[ Table[ Det[ Partition[ p[[i]], n]], {i, 1, (n^2)!}]]]) (* Robert G. Wilson v *)

Minor edits and a(6) from Hugo Pfoertner, Sep 08 2019

A180128 Maximal determinant of an n X n matrix whose elements are a permutation of the first n^2 prime numbers.

Original entry on oeis.org

1, 2, 29, 6640, 4868296, 5725998504, 11305600374272, 35954639671827328

Offset: 0

Hugo Pfoertner, Aug 11 2010

The terms a(5), a(6), a(7) were found by tabu search, with strong numerical evidence for the optimality of a(7).
A known lower bound for the next term a(8) is 154665569137423060000.
Upper bounds for higher terms can be found by the method described by O. Gasper, H. Pfoertner and M. Sigg, and are given in A180127, e.g., a(8) <= 154715716383037989022.
An improved lower bound is a(8) >= 154671943501236284416, provided in a private communication by Richard Gosiorovsky. - Hugo Pfoertner, Aug 27 2021

			a(2) = 29:
. 7 3
. 2 5
a(3) = 6640:
. 23 11  5
.  3 17 13
.  7  2 19
a(4) = 4868296:
. 53 11 23 13
. 17 47 29  3
.  7  5 43 37
. 19 31  2 41
a(5) = 5725998504
. 89 41 23  2 53
. 31 97 29 47 11
. 59 13 79 61  7
. 37 19  5 83 67
.  3 43 71 17 73
a(6) = 11305600374272:
. 137  73   7  89  83  13
.  79 139  67  19   3  97
. 101   5 149  61  37  53
.   2 109 103  71 113  11
.  59  29  41  17 131 127
.  23  47  43 151  31 107
a(7) = 35954639671827332:
. 227  71 173  43  83  29  73
. 151 163   5 181   2 103  89
.  31 223 139  61 137  97  13
.  23  47 157 211 109  19 131
. 113   7  67 127 167 199  17
.  53  79 149  37  11 193 179
. 101 107   3  41 191  59 197

Ortwin Gasper, Hugo Pfoertner and Markus Sigg, An Upper Bound for the Determinant of a Matrix with given Entry Sum and Square Sum JIPAM, Vol. 10, Iss. 3, Art. 63, 2008
Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO]
Index entries for sequences related to maximal determinants

Cf. A180127 [upper bounds for a(n)], A085000 [maximal determinants for matrix elements 1, ..., n^2].

Cf. A340923, A340924, A340925.

a(7) corrected, based on private communication from Richard Gosiorovsky by Hugo Pfoertner, Aug 27 2021

a(0)=1 prepended by Alois P. Heinz, Jan 19 2022

A088215 (1/36)*number of ways to express n as the determinant of a 3 X 3 matrix with elements 1..9.

Original entry on oeis.org

76, 32, 18, 30, 14, 47, 30, 25, 10, 41, 20, 42, 32, 25, 16, 36, 14, 31, 39, 28, 35, 39, 20, 22, 18, 33, 19, 45, 12, 21, 37, 26, 15, 41, 25, 37, 29, 27, 18, 34, 22, 24, 23, 24, 17, 48, 16, 16, 18, 15, 25, 35, 16, 21, 36, 43, 11, 30, 5, 18, 31, 17, 13, 28, 11, 42, 35, 24, 13, 35

Offset: 0

Hugo Pfoertner, Sep 23 2003

0 can be expressed in 36*76=2716 ways as the determinant of a 3 X 3 matrix which has elements 1..9. One such way is e.g. det ((1 2 3)(4 5 6)(7 8 9))=0. All numbers between -323 and +323 can be expressed by such a determinant. The first number not expressible is given by A088216.

Hugo Pfoertner, Table of n, a(n) for n = 0..412 [Gives all terms]
Hugo Pfoertner, Possible determinants of 3 X 3 matrix with elements 1..9.

Cf. A085000, A088214, A088216, A088217.

Cf. A136608 [analogous sequence for 4 X 4 matrices].

A350566 a(n) is the maximum permanent of an n X n matrix using the integers 1 to n^2.

Original entry on oeis.org

1, 1, 14, 947, 161388, 56558003, 36757837732

Offset: 0

Hugo Pfoertner at the suggestion of Stefano Spezia, Jan 21 2022

a(7) >= 38677620556961 corresponding to the matrix
  14, 25, 39,  3, 45,  2, 42
  32, 21, 10, 46,  5, 47,  8
  31, 20,  9, 48,  1, 49,  6
  44, 24, 18, 33, 13, 34, 15
  22, 29, 35, 12, 36, 11, 37
  16, 26, 38,  7, 43,  4, 40
  23, 41, 30, 19, 27, 17, 28 . - Robert Israel, Mar 19 2025
a(7) >= 38677691168324 corresponding to the matrix
   1,  4, 14, 25, 39, 42, 45
   5,  6, 16, 26, 38, 40, 43
  11, 12, 22, 29, 35, 36, 37
  17, 19, 23, 41, 30, 28, 27
  33, 34, 44, 24, 18, 15, 13
  48, 46, 32, 21, 10,  8,  3
  49, 47, 31, 20,  9,  7,  2. - Pontus von Brömssen, Mar 20 2025

			a(2) = 14:
  [2, 3;
   4, 1]
.
a(3) = 947:
  [3, 7, 6;
   9, 4, 1;
   2, 5, 8]
.
a(4) = 161388:
  [ 2,  3, 16,  6;
   11, 13,  4, 10;
    8,  9,  5, 15;
   14, 12,  1,  7]
.
a(5) = 56558003:
  [10,  2, 19, 25,  3;
   11,  5, 23, 20,  8;
   21, 14, 12,  9, 15;
   13, 24,  6,  1, 18;
   16, 17,  7,  4, 22]
.
a(6) = 36757837732:
  [32, 30,  3, 19, 23,  2;
    1,  5, 34, 14, 11, 36;
   17, 18, 15, 31, 22, 16;
   29, 28,  7, 20, 24,  6;
   26, 25, 10, 21, 27,  9;
    4,  8, 35, 13, 12, 33]

Carl-Erik Fröberg, On a combinatorial problem related to permanents, BIT 28 (1988), No. 3, 406-411.

Cf. A085000 (determinant), A350565 (minimum), A350858, A350859, A358487 (elements 0 to n^2-1).

from itertools import permutations
from sympy import Matrix
def A350566(n): return 1 if n == 0 else max(Matrix(n,n,p).per() for p in permutations(range(1,n**2+1))) # Chai Wah Wu, Jan 21 2022

cp's OEIS Frontend

A088214 Numbers less than the maximum possible determinant A085000(3)=412 not occurring as determinant of a 3 X 3 matrix with elements 1..9.

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A088237 Numbers less than the maximum possible determinant A085000(4)=40800 not occurring as determinant of a 4 X 4 matrix with elements 1..16.

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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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A088238 Numbers less than the maximum possible determinant A085000(5)=6839492 not occurring as determinant of a 5X5 matrix with elements 1..25.

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A301371 Maximum determinant of an n X n matrix with n copies of the numbers 1 .. n.

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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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A088217 Number of distinct values that can be assumed by the determinant of an n X n matrix whose entries are all permutations of the numbers 1..n^2.

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A180128 Maximal determinant of an n X n matrix whose elements are a permutation of the first n^2 prime numbers.

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A088215 (1/36)*number of ways to express n as the determinant of a 3 X 3 matrix with elements 1..9.

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A350566 a(n) is the maximum permanent of an n X n matrix using the integers 1 to n^2.

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