A088215 -id:A088215 - cp's OEIS frontend

A085000 Maximal determinant of an n X n matrix using the integers 1 to n^2.

1, 10, 412, 40800, 6839492, 1865999570, 762150368499

Offset: 1

Robert G. Wilson v, Jun 16 2003

Bounds for the next terms and the corresponding matrices are given by O. Gasper, H. Pfoertner and M. Sigg: 440960274696935 <= a(8) < 441077015225642, 346254605664223620 <= a(9) < 346335386150480625, 356944784622927045792 <= a(10) < 357017114947987625629. a(n) < sqrt(3*((n^5+n^4+n^3+n^2)/12)^n*(n^2+1)/(n+1)). - Hugo Pfoertner, Aug 15 2010
Improved lower bounds (private communication from Benjamin R. Buhrow, Dec 09 2019): a(8) >= 440970981670289, a(9) >= 346260899916111296. - Hugo Pfoertner, Jan 25 2021
Improved lower bound (private communication from Richard Gosiorovsky, Aug 18 2021): a(10) >= 356948996371054862392. - Hugo Pfoertner, Aug 24 2021

			The following 3 X 3 matrix is one of 36 whose determinant is 412 (there are also 36 whose determinant is -412):
   9 3 5
   4 8 1
   2 6 7
Results from a specially adapted hill-climbing algorithm strongly suggest that a(5) = 6839492. a(6) is at least 1862125166. Heuristically, a(n) is approximately 0.44*n^(2.06*n), suggesting that a(7) is close to 6.8*10^11. - Tim Paulden (timmy(AT)cantab.net), Sep 21 2003
a(5) confirmed by _Robert Israel_ and _Hugo Pfoertner_. A corresponding matrix is ((25 15 9 11 4) (7 24 14 3 17) (6 12 23 20 5) (10 13 2 22 19) (16 1 18 8 21) ). - _Hugo Pfoertner_, Sep 23 2003
a(6) found with FORTRAN program given at Pfoertner link. A corresponding matrix is ((36 24 21 17 5 8) ( 3 35 25 15 23 11) (13 7 34 16 10 31) (14 22 2 33 12 28) (20 4 19 29 32 6) (26 18 9 1 30 27) ). - _Hugo Pfoertner_, Sep 23 2003
a(7) is the determinant of the matrix ((46 42 15 2 27 24 18) (9 48 36 30 7 14 31) (39 11 44 34 13 29 5) (26 22 17 41 47 1 21) (20 8 40 6 33 23 45) (4 28 19 25 38 49 12) (32 16 3 37 10 35 43)). Although no proof for the optimality of a(7) is available, the results of an extensive computational search make the existence of a better solution extremely unlikely. A total of approximately 15 CPU years on SGI Origin 3000 and of 3.8 CPU years on SGI Altix 3000 computers was used for this result.

Sela Fried and Toufik Mansour, On the maximal sum of the entries of a matrix power, arXiv:2308.00348 [math.CO], 2023.
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.
Hugo Pfoertner, Maximal determinant of matrix with elements 1..n. FORTRAN program.
Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Square Matrix
Index entries for sequences related to maximal determinants

Cf. A088214, A088215, A088216, A088217, A088237, A180087 [upper bounds for a(n)], A180128, A221976, A301371, A301532, A301533, A309985, A325900, A350566.

Needs["DiscreteMath`Combinatorica`"]; n=3; n2=n^2; dMax=0; mMax={}; p=Range[n2]; Do[m=Partition[p, n]; d=Det[m]; If[d>dMax, dMax=d; mMax=m]; p=NextPermutation[p], {k, n2!}]; {dMax, mMax} (* T. D. Noe *)

vectomat(v)=my(n=sqrtint(#v));matrix(n,n,i,j,v[n*(i-1)+j])
a(n)=my(m,t,M); n*=n; for(k=0,(n-1)!-1, t=matdet(M=vectomat(numtoperm(n,k))); if(abs(t)>m, m=abs(t); print(t" "M)));m \\ Charles R Greathouse IV, Sep 13 2013

a(4) from Marsac Laurent (jko(AT)rox0r.net), Sep 15 2003
a(6) from Hugo Pfoertner, Sep 23 2003
Entry edited by N. J. A. Sloane, Nov 22 2006, to remove some erroneous entries. Further edits Nov 25 2006.
a(7) from Hugo Pfoertner, Jan 22 2008

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

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.

Original entry on oeis.org

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.

A136609 (1/(n!)^2) * number of ways to arrange the consecutive numbers 1...n^2 in an n X n matrix with determinant = 0.

Original entry on oeis.org

0, 0, 76, 14392910

Offset: 1

Hugo Pfoertner, Jan 21 2008

The computation of a(5) seems to be currently (Jan 2008) out of reach (compare with A088021(5)).

			a(1)=0 because det((1))/=0, a(2)=0, because the only possible determinants of a matrix with elements {1,2,3,4} are +-2, +-5 and +-10.

Cf. A001044, A046747, a(3)=A088215(0), a(4)=A136608(0), A221976.

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.

A180345 Lexicographically ordered 3X3 matrices containing numbers 1..9 with maximal determinant = 412.

Original entry on oeis.org

148726593, 157836492, 175429863, 184539762, 249715683, 267935481, 276418953, 294638751, 359814672, 368924571, 386517942, 395627841, 418953276, 429863175, 481267935, 492157836, 517942386, 539762184

Offset: 1

Zak Seidov, Jan 18 2011

The matrices are presented here as 9-digit decimal numbers, one digit per entry in the matrix.

There are exactly 36 such matrices: 148726593, 157836492, 175429863, 184539762, 249715683, 267935481, 276418953, 294638751, 359814672, 368924571, 386517942, 395627841, 418953276, 429863175, 481267935, 492157836, 517942386, 539762184, 571368924, 593148726, 627841395, 638751294, 672359814, 683249715, 715683249, 726593148, 751294638, 762184539, 814672359, 836492157, 841395627, 863175429, 924571368, 935481267, 942386517, 953276418.

			148726593 => {{1,4,8},{7,2,6},{5,9,3}}:
1 4 8
7 2 6
5 9 3
1*(2*3-9*6)-4(7*3-5*6)+8*(7*9-5*2)=412.

Cf. A085000 Maximal determinant of an n X n matrix using the integers 1 to n^2.

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

A372708 a(n) is the smallest number k that is the concatenation of the elements of a 3 X 3 matrix whose determinant is n and whose elements are a permutation of the numbers 1 through 9; a(n) = -1 if no such number k exists.

Original entry on oeis.org

123456789, 123469857, 123467589, 123467895, 123458769, 123469578, 123589476, 123457689, 123748569, 123456798, 123469587, 123469875, 123458967, 123457986, 123469785, 123457698, 123548769, 123689574, 123546789, 123457896, 123569487, 123458697, 123547689, 123649758, 123567498

Offset: 0

Jean-Marc Rebert, May 11 2024

The determinant of a 3 X 3 matrix whose elements are a permutation of the numbers 1..9 cannot exceed 412, so this sequence is finite.

			a(0) = 123456789 because it is the smallest number that can be formed by concatenating the elements of a 3 X 3 matrix whose determinant is 0 and whose elements are a permutation of the numbers 1..9. The matrix is [1 2 3] [4 5 6] [7 8 9].

Cf. A085000, A088214, A088215, A221976.

a[n_] := a[n] = Module[{mat}, mat = Select[Partition[#, 3] & /@ Permutations[Range[1, 9]], Det[#] == n &]; If[Length[mat] > 0, First[Sort[ToExpression[StringJoin[Riffle[ToString /@ Flatten[#], ""]]] & /@ mat]], 0]];
Monitor[(* Do not use Monitor[] if using Wolfram Cloud, otherwise memory issues may occur *)Table[a[n], {n, 0, 24}], {n, Table[a[m], {m, 0, n - 1}]}] (* Robert P. P. McKone, May 11 2024 *)

from sympy import Matrix
from itertools import permutations
adict = dict()
for p in permutations(range(1, 10)):
    v = Matrix(3, 3, p).det()
    if v not in adict:
        adict[v] = int("".join(map(str, p)))
afull = [adict[v] if v in adict else -1 for v in range(max(adict)+1)]
print(afull) # Michael S. Branicky, May 11 2024

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A085000 Maximal determinant of an n X n matrix using the integers 1 to n^2.

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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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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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A136609 (1/(n!)^2) * number of ways to arrange the consecutive numbers 1...n^2 in an n X n matrix with determinant = 0.

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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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A180345 Lexicographically ordered 3X3 matrices containing numbers 1..9 with maximal determinant = 412.

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A372708 a(n) is the smallest number k that is the concatenation of the elements of a 3 X 3 matrix whose determinant is n and whose elements are a permutation of the numbers 1 through 9; a(n) = -1 if no such number k exists.

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