A085000 -id:A085000 - cp's OEIS frontend

A350565 a(n) is the minimum permanent of an n X n matrix using the integers 1 to n^2.

1, 1, 10, 436, 52197, 13300936, 6192060119

Offset: 0

Stefano Spezia and Hugo Pfoertner, Jan 20 2022

a(7) <= 4755379618016 from the matrix
  [ 1,  2,  3,  4,  5,  6,  7;
    8, 14, 19, 23, 29, 33, 36;
    9, 15, 21, 26, 32, 37, 41;
   10, 16, 22, 28, 35, 40, 44;
   11, 17, 24, 30, 38, 43, 46;
   12, 18, 25, 31, 39, 45, 48;
   13, 20, 27, 34, 42, 47, 49]. - Pontus von Brömssen, Aug 30 2025

			a(2) = 10:
  [1, 3;
   2, 4]
.
a(3) = 436:
  [1, 3, 2;
   4, 8, 6;
   5, 9, 7]
.
a(4) = 52197:
  [1,  2,  4,  3;
   6,  9, 15, 12;
   5,  8, 13, 11;
   7, 10, 16, 14]
.
a(5) = 13300936:
  [16,  8, 24, 21, 12;
   18,  9, 25, 23, 13;
    3,  1,  5,  4,  2;
   14,  6, 20, 17, 10;
   15,  7, 22, 19, 11]
.
a(6) = 6192060119:
  [36, 35, 33, 31, 27,  6;
   11, 10,  9,  8,  7,  1;
   34, 32, 30, 28, 25,  5;
   22, 21, 19, 18, 16,  3;
   29, 26, 24, 23, 20,  4;
   17, 15, 14, 13, 12,  2]

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

Cf. A085000, A350566 (maximum), A350858, A350859, A358486 (elements 0 to n^2-1).

from itertools import permutations
from sympy import Matrix
def A350565(n): return 1 if n == 0 else min(Matrix(n,n,p).per() for p in permutations(range(1,n**2+1))) # Chai Wah Wu, Jan 21 2022

A097399 Maximum of the determinant over all permutations of the entries of a 3 X 3 matrix which are consecutive integers in the range (n-4,n+4).

Original entry on oeis.org

86, 104, 172, 252, 332, 412, 492, 572, 652, 732, 812, 892, 972, 1053, 1134, 1215, 1296, 1377, 1458, 1539, 1620, 1701, 1782, 1863, 1944, 2025, 2106, 2187, 2268, 2349, 2430, 2511, 2592, 2673, 2754, 2835, 2916, 2997, 3078, 3159, 3240, 3321, 3402, 3483, 3564

Offset: 0

Hugo Pfoertner, Aug 19 2004

nonn

			a(0)=86 because the maximal determinant that can achieved using the consecutive integers -4,-3,-2,-1,0,1,2,3,4 as matrix elements of a 3 X 3 matrix is det((-4,-3,0),(1,-1,4),(-2,3,2))=86. Another example for a(5)=412 is given in A085000.

Cf. A097400 = corresponding number of different determinants, A097401, A097693 = maximum of determinant if distinct matrix elements are selected from given range, a(5)=A085000(3) maximal determinant with elements (1..n^2).

Join[{86,104,172,252,332,412,492,572,652,732,812,892},LinearRecurrence[ {2,-1},{972,1053},40]] (* or *) Table[ Det[ Partition[ #,3]]&/@ Permutations[ Range[n-4,n+4]]//Max,{n,0,45}] (* Harvey P. Dale, Jan 14 2015 *)

G.f.: (x^13+12*x^3+50*x^2-68*x+86) / (x-1)^2. [Colin Barker, Dec 13 2012] [I suspect this is merely a conjecture. - N. J. A. Sloane, Jun 09 2018]

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

Original entry on oeis.org

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).

A351609 Maximal absolute value of the determinant of an n X n symmetric matrix using the integers 1 to n*(n + 1)/2.

Original entry on oeis.org

1, 1, 7, 152, 7113, 745285, 94974369

Offset: 0

Stefano Spezia, Feb 14 2022

Upper bounds for the next terms can be found by considering all possibilities of choosing matrix entries on the diagonal and applying Gasper's determinant theorem (see references in A085000): a(7) <= 22475584128, a(8) <= 6634478203404, a(9) <= 2647044512044258. - Hugo Pfoertner, Feb 18 2022

			a(3) = 152:
   2    4    6
   4    5    1
   6    1    3
a(4) = 7113:
   2    6    8    9
   6    5   10    1
   8   10    3    4
   9    1    4    7

Cf. A000217, A351147, A351148, A351153.

Cf. A085000, A180128, A350931, A350933, A350954, A350956.

a(n) = max(abs(A351147(n)), A351148(n)). - Hugo Pfoertner, Feb 16 2022

a(5)-a(6) from Hugo Pfoertner, Feb 16 2022

A097694 Largest achievable determinant of a 4 X 4 matrix whose elements are 16 distinct nonnegative integers chosen from the range 0...n.

Original entry on oeis.org

36000, 50736, 69828, 94092, 124699, 162604, 208697, 264094, 329983, 407624, 498349, 603562

Offset: 15

Hugo Pfoertner, Aug 24 2004

			a(18)=94092 because no 4 X 4 matrix b(j,k) with distinct elements 0<=b(j,k)<=18,j=1..4,k=1..4 can be built that has a larger determinant than
det((18,6,2,10),(3,17,4,13),(11,12,16,0),(5,1,14,15))=94092.

Other maximal 4 X 4 determinants: Cf. A097696: 4 X 4 matrix filled with consecutive integers, A097695: 4 X 4 matrix filled with integers from -n...n, A097399, A097401, A097693: corresponding sequences for 3 X 3 matrices, A085000: n X n matrix filled with consecutive integers 1...n^2.

For n<=18 an optimal choice and arrangement is of the following form det((n, 6, 2, n-8), (3, n-1, 4, n-5), (n-7, n-6, n-2, 0), (5, 1, n-4, n-3))= 3*n^4-62*n^3+543*n^2-2128*n+3120. For n>18 up to the limit investigated (n<=26) one choice maximizing the determinant is det((n, 6, 2, n-8), (3, n-1, 4, n-5), (n-7, n-6, n-3, 0), (5, 1, n-2, n-4))= 3*n^4-60*n^3+491*n^2-1821*n+2624. In both cases there are 575 other equivalent arrangements corresponding to permutations of rows and columns.

A097696 Largest achievable determinant of a 4 X 4 matrix whose elements are the 16 consecutive integers n-15,...,n.

Original entry on oeis.org

7343, 8784, 12065, 16800, 21600, 26400, 31200, 36000, 40800, 45600, 50400, 55200, 60000, 64800, 69600, 74400, 79200, 84000, 88800, 93600, 98400, 103200, 108000, 112800, 117600, 122400, 127200, 132000, 136800, 141600, 146400, 151200, 156000

Offset: 8

Hugo Pfoertner, Aug 25 2004

nonn

Other maximal 4 X 4 determinants: Cf. A097694: 4 X 4 matrix filled with integers from 0...n, A097695: 4 X 4 matrix filled with integers from -n...n. A097399, A097401, A097693: corresponding sequences for 3 X 3 matrices. a(16)=A085000(4).

For n>10 an arrangement maximizing the determinant is of the following form: det((n, n-9, n-13, n-8), (n-12, n-1, n-11, n-5), (n-7, n-6, n-2, n-15), (n-10, n-14, n-4, n-3)) =2400*(2*n-15). a(n)=a(15-n) for n<8.

Empirical G.f.: x^8*(65*x^4+1454*x^3+1840*x^2-5902*x+7343) / (x-1)^2. [Colin Barker, Jan 10 2013]

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.

A180087 Upper bound for the determinant of a matrix whose entries are a permutation of 1, ..., n^2.

Original entry on oeis.org

1, 11, 450, 41021, 6865625, 1867994210, 762539814814, 441077015225642, 346335386150480625, 357017114947987625629, 470379650542113331346272, 774869480550211708169959725, 1566955892015559322525350178004

Offset: 1

Hugo Pfoertner, Aug 09 2010

nonn

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.

Rainer Rosenthal, Table of n, a(n) for n = 1..191
O. Gasper, H. Pfoertner and M. 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], 2018.

a(n) is an upper bound for A085000(n).

a(n) = floor(sqrt(3*((n^5+n^4+n^3+n^2)/12)^n*(n^2+1)/(n+1))).

A301532 Maximum determinant of an n X n matrix with entries 1, 1/2, .., 1/n^2; numerator.

Original entry on oeis.org

1, 5, 2027, 12976897, 450724396028209, 13238878814817907394909, 280849389948155488261365087763753, 132758211671968916518163154756197108235468015014261

Offset: 1

Hugo Pfoertner, Mar 23 2018

The maximum determinant achievable by arranging the fractions 1/1, 1/2, 1/3, ..., 1/n^2 as matrix entries is provided as fraction a(n) / A301533(n).

			a(3) = 2027, because no matrix with a greater determinant can be found than
  (1/1 1/7 1/5)
  (1/4 1/2 1/9)
  (1/8 1/6 1/3),
which has the determinant 2027/15120. A301533(3) = 15120.

Hugo Pfoertner, Illustration of maximum determinant value A301532/A301533.

Cf. A085000, A301371, A301533 (corresponding denominators)

A351147 a(n) is the minimal determinant of a symmetric n X n matrix using the integers 1 to n*(n+1)/2.

Original entry on oeis.org

1, -7, -152, -6276, -550010, -94974369

Offset: 1

Hugo Pfoertner at the suggestion of Markus Sigg, Feb 15 2022

			a(2) = -7:
  [1, 3;
   3, 2]
.
a(3) = -152:
  [5, 4, 1;
   4, 2, 6;
   1, 6, 3]
.
a(4) = -6276:
  [9,  5,  2,  4;
   5,  6, 10,  1;
   2, 10,  3,  7;
   4,  1,  7,  8]
.
a(5) = -550010:
  [ 3, 14,  7, 13,  9;
   14,  8,  1, 12,  4;
    7,  1,  6, 10, 15;
   13, 12, 10,  2, 11;
    9,  4, 15, 11,  5]
.
a(6) = -94974369:
  [ 5, 14,  7, 11, 21, 17;
   14, 12, 19,  3, 16,  1;
    7, 19, 13, 18,  2,  6;
   11,  3, 18, 10,  4, 20;
   21, 16,  2,  4,  9, 15;
   17,  1,  6, 20, 15,  8]

Cf. A000217, A085000, A351148, A351609.

cp's OEIS Frontend

A350565 a(n) is the minimum permanent of an n X n matrix using the integers 1 to n^2.

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A097399 Maximum of the determinant over all permutations of the entries of a 3 X 3 matrix which are consecutive integers in the range (n-4,n+4).

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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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A351609 Maximal absolute value of the determinant of an n X n symmetric matrix using the integers 1 to n*(n + 1)/2.

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A097694 Largest achievable determinant of a 4 X 4 matrix whose elements are 16 distinct nonnegative integers chosen from the range 0...n.

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A097696 Largest achievable determinant of a 4 X 4 matrix whose elements are the 16 consecutive integers n-15,...,n.

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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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A180087 Upper bound for the determinant of a matrix whose entries are a permutation of 1, ..., n^2.

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A301532 Maximum determinant of an n X n matrix with entries 1, 1/2, .., 1/n^2; numerator.

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A351147 a(n) is the minimal determinant of a symmetric n X n matrix using the integers 1 to n*(n+1)/2.

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