cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A358323 a(n) is the minimal determinant of an n X n symmetric Toeplitz matrix using the integers 0 to n - 1.

Original entry on oeis.org

1, 0, -1, -7, -60, -1210, -34020, -607332, -30448441, -1093612784, -55400732937, -2471079070511, -197500419383964
Offset: 0

Views

Author

Stefano Spezia, Nov 09 2022

Keywords

Examples

			a(3) = -7:
    [1, 2, 0;
     2, 1, 2;
     0, 2, 1]
a(4) = -60:
    [2, 3, 0, 1;
     3, 2, 3, 0;
     0, 3, 2, 3;
     1, 0, 3, 2]
a(5) = -1210:
    [4, 3, 0, 2, 1;
     3, 4, 3, 0, 2;
     0, 3, 4, 3, 0;
     2, 0, 3, 4, 3;
     1, 2, 0, 3, 4]

Links

Crossrefs

Cf. A350953.
Cf. A358324 (maximal), A358325 (minimal nonzero absolute value), A358326 (minimal permanent), A358327 (maximal permanent).

Programs

  • Mathematica
    Join[{1}, Table[Min[Table[Det[ToeplitzMatrix[Part[Permutations[Join[{0}, Range[n-1]]], i]]],{i,n!}]],{n,9}]]

Extensions

a(10)-a(12) from Lucas A. Brown, Nov 16 2022

A358324 a(n) is the maximal determinant of an n X n symmetric Toeplitz matrix using the integers 0 to n - 1.

Original entry on oeis.org

1, 0, 1, 8, 63, 2090, 36875, 1123653, 34292912, 1246207300, 53002204560, 2418538080316, 215120941720912
Offset: 0

Views

Author

Stefano Spezia, Nov 09 2022

Keywords

Examples

			a(3) = 8:
    [0, 2, 1;
     2, 0, 2;
     1, 2, 0]
a(4) = 63:
    [1, 3, 2, 0;
     3, 1, 3, 2;
     2, 3, 1, 3;
     0, 2, 3, 1]
a(5) = 2090:
    [2, 4, 0, 1, 3;
     4, 2, 4, 0, 1;
     0, 4, 2, 4, 0;
     1, 0, 4, 2, 4;
     3, 1, 0, 4, 2]

Links

Crossrefs

Cf. A350954.
Cf. A358323 (minimal), A358325 (minimal nonzero absolute value), A358326 (minimal permanent), A358327 (maximal permanent).

Programs

  • Mathematica
    Join[{1}, Table[Max[Table[Det[ToeplitzMatrix[Part[Permutations[Join[{0}, Range[n-1]]], i]]],{i,n!}]],{n,9}]]

Extensions

a(10)-a(12) from Lucas A. Brown, Nov 16 2022

A358326 a(n) is the minimal permanent of an n X n symmetric Toeplitz matrix using the integers 0 to n - 1.

Original entry on oeis.org

1, 0, 1, 4, 34, 744, 17585, 688202, 33248174, 2144597292, 169696358796, 16521881847592
Offset: 0

Views

Author

Stefano Spezia, Nov 09 2022

Keywords

Examples

			a(3) = 4:
    [0, 1, 2;
     1, 0, 1;
     2, 1, 0]
a(4) = 34:
    [1, 0, 2, 3;
     0, 1, 0, 2;
     2, 0, 1, 0;
     3, 2, 0, 1]
a(5) = 744:
    [1, 0, 2, 3, 4;
     0, 1, 0, 2, 3;
     2, 0, 1, 0, 2;
     3, 2, 0, 1, 0;
     4, 3, 2, 0, 1]

Links

Crossrefs

Cf. A351019.
Cf. A358323 (minimal determinant), A358324 (maximal determinant), A358327 (maximal).

Programs

  • Mathematica
    Join[{1}, Table[Min[Table[Permanent[ToeplitzMatrix[Part[Permutations[Join[{0}, Range[n-1]]], i]]],{i,n!}]],{n,9}]]

Extensions

a(10) and a(11) from Lucas A. Brown, Nov 16 2022

A369835 a(n) is the number of distinct values of the permanent of an n X n symmetric Toeplitz matrix using the integers 0 to n-1.

Original entry on oeis.org

1, 1, 1, 6, 23, 119, 718, 5038, 40320, 362879
Offset: 0

Views

Author

Stefano Spezia, Feb 03 2024

Keywords

Links

Crossrefs

Cf. A000142, A358326, A358327.
Cf. A369830, A369831, A369832, A369833, A369834.

Programs

  • Mathematica
    a[n_] := CountDistinct[Table[Permanent[ToeplitzMatrix[Part[Permutations[Join[{0}, Range[n - 1]]], i]]], {i, n !}]]; Join[{1}, Array[a,9]]
  • Python
    from itertools import permutations
from sympy import Matrix
def A369835(n): return len({Matrix([p[i:0:-1]+p[:n-i] for i in range(n)]).per() for p in permutations(range(n))}) # Chai Wah Wu, Feb 11 2024

Formula

a(n) <= A000142(n).

A358325 a(n) is the minimal absolute value of determinant of a nonsingular n X n symmetric Toeplitz matrix using the integers 0 to n - 1.

Original entry on oeis.org

1, 3, 12, 2, 11, 10, 5, 4, 1, 4, 1
Offset: 2

Views

Author

Stefano Spezia, Nov 09 2022

Keywords

Examples

			a(3) = 3:
    [1, 0, 2;
     0, 1, 0;
     2, 0, 1]
a(4) = 12:
    [3, 2, 1, 0;
     2, 3, 2, 1;
     1, 2, 3, 2;
     0, 1, 2, 3]
a(5) = 2:
    [0, 4, 1, 2, 3;
     4, 0, 4, 1, 2;
     1, 4, 0, 4, 1;
     2, 1, 4, 0, 4;
     3, 2, 1, 4, 0]

Links

Crossrefs

Cf. A356865.
Cf. A358323 (minimal signed), A358324 (maximal signed), A358326 (minimal permanent), A358327 (maximal permanent).

Programs

  • Mathematica
    Table[Min[Select[Table[Abs[Det[ToeplitzMatrix[Part[Permutations[Join[{0}, Range[n-1]]], i]]]], {i, n!}], Positive]], {n, 2, 9}]  (* Corrected by Stefano Spezia, Nov 17 2022 *)

Extensions

a(3), a(7), and a(10)-a(12) from Lucas A. Brown, Nov 16 2022

A374283 a(n) is the maximal permanent of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

Original entry on oeis.org

1, 0, 1, 8, 256, 9978, 600052, 49036950, 5286564352, 725724599636
Offset: 0

Views

Author

Stefano Spezia, Jul 02 2024

Keywords

Examples

			a(5) = 9978:
  [0, 4, 3, 2, 1]
  [4, 0, 4, 3, 2]
  [3, 4, 0, 4, 3]
  [2, 3, 4, 0, 4]
  [1, 2, 3, 4, 0]

Links

Crossrefs

Cf. A085807 (minimal), A358327.
Cf. A374279, A374280, A374281, A374282.

Programs

  • Mathematica
    a[0]=1; a[n_]:=Max[Table[Permanent[ToeplitzMatrix[Join[{0}, Part[Permutations[Range[n - 1]], i]]]], {i, (n-1)!}]]; Array[a, 11, 0]

A364790 Triangle read by rows: T(n, k) is the number of n X n symmetric Toeplitz matrices of rank k using all the integers 0, 1, 2, ..., n-1.

Original entry on oeis.org

1, 0, 2, 0, 0, 6, 0, 0, 1, 23, 0, 0, 0, 0, 120, 0, 0, 0, 0, 2, 718, 0, 0, 0, 0, 4, 31, 5005, 0, 0, 0, 0, 0, 2, 44, 40274, 0, 0, 0, 0, 0, 0, 4, 284, 362592, 0, 0, 0, 0, 0, 0, 0, 111, 769, 3627920, 0, 0, 0, 0, 0, 0, 2, 14, 244, 7056, 39909484, 0, 0, 0, 0, 0, 0, 0, 4, 64, 742, 9667, 478991123
Offset: 1

Views

Author

Stefano Spezia, Aug 08 2023

Keywords

Examples

			The triangle begins:
  1;
  0, 2;
  0, 0, 6;
  0, 0, 1, 23;
  0, 0, 0,  0, 120;
  0, 0, 0,  0,   2, 718;
  0, 0, 0,  0,   4,  31, 5005;
  0, 0, 0,  0,   0,   2,   44, 40274;
  0, 0, 0,  0,   0,   0,    4,   284, 362592;
  ...

Links

Crossrefs

Cf. A000142 (row sums), A358323 (minimal determinant), A358324 (maximal determinant), A358326 (minimal permanent), A358327 (maximal permanent), A364791 (right diagonal).

Programs

  • Mathematica
    T[n_, k_]:= Count[Table[MatrixRank[ToeplitzMatrix[Part[Permutations[Join[{0},Range[n-1]]], i]]], {i, n!}], k]; Join[{1},Table[T[n, k], {n,2,9}, {k, n}]]//Flatten
  • PARI
    MkMat(v)={matrix(#v, #v, i, j, v[1+abs(i-j)])}
row(n)={if(n==1, [1], my(f=vector(n)); forperm(vector(n, i, i-1), v, f[matrank(MkMat(v))]++); f)} \\ Andrew Howroyd, Jan 07 2024

Extensions

Terms a(46) and beyond from Andrew Howroyd, Jan 07 2024

A364791 a(n) is the number of n X n nonsingular symmetric Toeplitz matrices using all the integers 0, 1, 2, ..., n-1.

Original entry on oeis.org

1, 2, 6, 23, 120, 718, 5005, 40274, 362592, 3627920, 39909484, 478991123
Offset: 1

Views

Author

Stefano Spezia, Aug 08 2023

Keywords

Links

Crossrefs

Right diagonal of A364790.
Cf. A358323 (minimal determinant), A358324 (maximal determinant), A358326 (minimal permanent), A358327 (maximal permanent).

Programs

  • Mathematica
    a[n_]:=Count[Table[MatrixRank[ToeplitzMatrix[Part[Permutations[Join[{0}, Range[n - 1]]], i]]], {i, n!}], n]; Join[{1}, Array[a, 9, 2]] // Flatten

Extensions

a(10)-a(12) from Andrew Howroyd, Jan 07 2024
Showing 1-8 of 8 results.

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