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.

A359617 a(n) is the maximal permanent of an n X n Hermitian Toeplitz matrix using all the integers 1, 2, ..., n and with all off-diagonal elements purely imaginary.

Original entry on oeis.org

1, 1, 5, 54, 980, 26775, 1061841, 56647472, 4103545288, 367479636012
Offset: 0

Views

Author

Stefano Spezia, Jan 07 2023

Keywords

Examples

			a(4) = 980:
  [   4,  3*i,  2*i,   i;
   -3*i,    4,  3*i, 2*i;
   -2*i, -3*i,    4, 3*i;
     -i, -2*i, -3*i,   4 ]

Links

Crossrefs

Cf. A351020, A359560, A359562.
Cf. A359614 (minimal determinant), A359615 (maximal determinant), A359616 (minimal).

Programs

  • Mathematica
    a={1}; For[n=1, n<=7, n++, mx=-Infinity; For[d=1, d<=n, d++, For[i=1, i<=(n-1)!, i++, If[(t=Permanent[ToeplitzMatrix[Join[{d}, I Part[Permutations[Drop[Range[n], {d}]], i]]]])>mx, mx=t]]]; AppendTo[a, mx]]; a
  • Python
    from itertools import permutations
from sympy import Matrix, I
def A359617(n): return max(Matrix(n,n,[(d[i-j] if i>j else -d[j-i]) if i!=j else d[0]*I for i in range(n) for j in range(n)]).per()*(1,-I,-1,I)[n&3] for d in permutations(range(1,n+1))) if n else 1 # Chai Wah Wu, Jan 25 2023

A351019 Minimal permanent of an n X n symmetric Toeplitz matrix using the integers 1 to n.

Original entry on oeis.org

1, 1, 5, 36, 480, 9991, 296913, 12099604, 637590728, 43090005714, 3550491371994, 359557627057876
Offset: 0

Views

Author

Stefano Spezia, Jan 29 2022

Keywords

Examples

			a(3) = 36:
    2    1    3
    1    2    1
    3    1    2
a(4) = 480:
    2    1    3    4
    1    2    1    3
    3    1    2    1
    4    3    1    2
a(5) = 9991:
    3    1    2    4    5
    1    3    1    2    4
    2    1    3    1    2
    4    2    1    3    1
    5    4    2    1    3

Links

Crossrefs

Cf. A204235, A307783, A350937, A351020 (maximal).

Programs

  • Python
    from itertools import permutations
from sympy import Matrix
def A351019(n): return 1 if n == 0 else min(Matrix([p[i:0:-1]+p[0:n-i] for i in range(n)]).per() for p in permutations(range(1,n+1))) # Chai Wah Wu, Jan 31 2022

Extensions

a(9) from Alois P. Heinz, Jan 31 2022
a(10)-a(11) from Lucas A. Brown, Sep 06 2022

A358327 a(n) is the maximal permanent of an n X n symmetric Toeplitz matrix using the integers 0 to n - 1.

Original entry on oeis.org

1, 0, 1, 12, 304, 12696, 778785, 64118596, 7014698888, 965862895732, 166105870928994, 34460169208369298
Offset: 0

Views

Author

Stefano Spezia, Nov 09 2022

Keywords

Examples

			a(3) = 12:
    [2, 1, 0;
     1, 2, 1;
     0, 1, 2]
a(4) = 304:
    [2, 3, 1, 0;
     3, 2, 3, 1;
     1, 3, 2, 3;
     0, 1, 3, 2]
a(5) = 12696:
    [3, 4, 2, 1, 0;
     4, 3, 4, 2, 1;
     2, 4, 3, 4, 2;
     1, 2, 4, 3, 4;
     0, 1, 2, 4, 3]

Links

Crossrefs

Cf. A351020.
Cf. A358323 (minimal determinant), A358324 (maximal determinant), A358326 (minimal).

Programs

  • Mathematica
    Join[{1}, Table[Max[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

A369831 a(n) is the number of distinct values of the permanent of an n X n symmetric Toeplitz matrix using the integers 1 to n.

Original entry on oeis.org

1, 1, 1, 6, 23, 120, 720, 5040, 40320, 362880
Offset: 0

Views

Author

Stefano Spezia, Feb 03 2024

Keywords

Links

Crossrefs

Cf. A000142, A351019, A351020.
Cf. A369830, A369832, A369833, A369834, A369835.

Programs

  • Mathematica
    a[n_] := CountDistinct[Table[Permanent[ToeplitzMatrix[Part[Permutations[Range[n]],i]]], {i, n!}]]; Join[{1}, Array[a,9]]
  • Python
    from itertools import permutations
from sympy import Matrix
def A369831(n): return len({Matrix([p[i:0:-1]+p[:n-i] for i in range(n)]).per() for p in permutations(range(1,n+1))}) # Chai Wah Wu, Feb 12 2024

Formula

a(n) <= A000142(n).
Conjectured e.g.f.: 1/(1 - x) - x^2/2 - x^4/24.

A351611 Maximal permanent of an n X n symmetric matrix using the integers 1 to n*(n + 1)/2.

Original entry on oeis.org

1, 1, 11, 420, 41451, 7985639, 2779152652
Offset: 0

Views

Author

Stefano Spezia, Feb 14 2022

Keywords

Examples

			a(3) = 420:
    1    5    6
    5    3    4
    6    4    2
a(4) = 41451:
    1    5    8   10
    5    4    9    7
    8    9    3    6
   10    7    6    2

Crossrefs

Cf. A000217, A351153, A351610 (minimal).
Cf. A350566, A350859, A350938, A350940, A351020, A351022.

Extensions

a(5)-a(6) from Hugo Pfoertner, Feb 15 2022

A364230 Triangle read by rows: T(n, k) is the number of n X n symmetric Toeplitz matrices of rank k using all the integers 1, 2, ..., n.

Original entry on oeis.org

1, 0, 2, 0, 0, 6, 0, 0, 0, 24, 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, 272, 362604, 0, 0, 0, 0, 0, 0, 0, 111, 774, 3627915, 0, 0, 0, 0, 0, 0, 2, 14, 244, 6974, 39909566, 0, 0, 0, 0, 0, 0, 0, 4, 64, 743, 9533, 478991256
Offset: 1

Views

Author

Stefano Spezia, Jul 14 2023

Keywords

Examples

			The triangle begins:
  1;
  0, 2;
  0, 0, 6;
  0, 0, 0, 24;
  0, 0, 0,  0, 120;
  0, 0, 0,  0,   2, 718;
  0, 0, 0,  0,   4,  31, 5005;
  ...

Links

Crossrefs

Cf. A000142 (row sums), A350953 (minimal determinant), A350954 (maximal determinant), A351019 (minimal permanent), A351020 (maximal permanent), A356865 (minimal nonzero absolute value determinant), A364231 (right diagonal).

Programs

  • Mathematica
    T[n_,k_]:= Count[Table[MatrixRank[ToeplitzMatrix[Part[Permutations[Range[n]], i]]],{i,n!}],k]; Table[T[n,k],{n,8},{k,n}]//Flatten
  • PARI
    MkMat(v)={matrix(#v, #v, i, j, v[1+abs(i-j)])}
row(n)={my(f=vector(n)); forperm(vector(n,i,i), v, f[matrank(MkMat(v))]++); f} \\ Andrew Howroyd, Dec 30 2023

Extensions

Terms a(46) and beyond from Andrew Howroyd, Dec 30 2023

A364231 a(n) is the number of n X n nonsingular symmetric Toeplitz matrices using all the integers 1, 2, ..., n.

Original entry on oeis.org

1, 2, 6, 24, 120, 718, 5005, 40274, 362604, 3627915, 39909566, 478991256
Offset: 1

Views

Author

Stefano Spezia, Jul 14 2023

Keywords

Links

Crossrefs

Right diagonal of A364230.
Cf. A350953 (minimal determinant), A350954 (maximal determinant), A351019 (minimal permanent), A351020 (maximal permanent), A356865 (minimal nonzero absolute value determinant).

Programs

  • Mathematica
    a[n_]:=Count[Table[MatrixRank[ToeplitzMatrix[Part[Permutations[Range[n]],i]]],{i,n!}],n]; Array[a,8]

Extensions

a(10)-a(12) from Andrew Howroyd, Dec 30 2023

A374278 a(n) is the maximal permanent of an n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

Original entry on oeis.org

1, 1, 2, 18, 389, 14284, 798322, 62490160, 6519511313, 873036867840, 145856387327074
Offset: 0

Views

Author

Stefano Spezia, Jul 02 2024

Keywords

Examples

			a(5) = 14284:
  [1, 4, 3, 2, 1]
  [4, 1, 4, 3, 2]
  [3, 4, 1, 4, 3]
  [2, 3, 4, 1, 4]
  [1, 2, 3, 4, 1]

Links

Crossrefs

Cf. A351020, A374139, A374140 (minimal).
Cf. A374239, A374240, A374241, A374242.

Programs

  • Mathematica
    a[0]=1; a[n_]:=Max[Table[Permanent[ToeplitzMatrix[Join[{1}, Part[Permutations[Range[n - 1]], i]]]], {i, (n-1)!}]]; Array[a, 11, 0]
Showing 1-8 of 8 results.

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