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.

A359615 a(n) is the maximal determinant 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, 3, 9, 512, 9195, 242931, 7459494, 524426191, 17012915860, 773407040859
Offset: 0

Views

Author

Stefano Spezia, Jan 07 2023

Keywords

Examples

			a(4) = 512:
  [   1,  4*i,  2*i, 3*i;
   -4*i,    1,  4*i, 2*i;
   -2*i, -4*i,    1, 4*i;
   -3*i, -2*i, -4*i,   1 ]

Links

Crossrefs

Cf. A350954, A359559, A359561.
Cf. A359614 (minimal), A359616 (minimal permanent), A359617 (maximal permanent).

Programs

  • Mathematica
    a={1}; For[n=1, n<=8, n++, mx=-Infinity; For[d=1, d<=n, d++, For[i=1, i<=(n-1)!, i++, If[(t=Det[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 A359615(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)]).det()*(1,-I,-1,I)[n&3] for d in permutations(range(1,n+1))) # Chai Wah Wu, Jan 25 2023

A359616 a(n) is the minimal 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, 18, 245, 2249, 57213, 947177, 50431724, 1282453618
Offset: 0

Views

Author

Stefano Spezia, Jan 07 2023

Keywords

Examples

			a(4) = 245:
  [   1,  3*i,  2*i, 4*i;
   -3*i,    1,  3*i, 2*i;
   -2*i, -3*i,    1, 3*i;
   -4*i, -2*i, -3*i,   1 ]

Links

Crossrefs

Cf. A351019, A359560, A359562.
Cf. A359614 (minimal determinant), A359615 (maximal determinant), A359617 (maximal).

Programs

  • Mathematica
    a={1}; For[n=1, n<=7, n++, mn=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]]]])
  • Python
    from itertools import permutations
from sympy import Matrix, I
def A359616(n): return min(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

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

A359559 a(n) is the determinant of an n X n Hermitian Toeplitz matrix whose first row consists of 1, 2*i, ..., n*i, where i denotes the imaginary unit.

Original entry on oeis.org

1, 1, -3, -16, -36, -40, 20, 184, 400, 432, -112, -1472, -3136, -3328, 576, 9856, 20736, 21760, -2816, -59392, -123904, -129024, 13312, 333824, 692224, 716800, -61440, -1785856, -3686400, -3801088, 278528, 9207808, 18939904, 19464192, -1245184, -46137344, -94633984
Offset: 0

Views

Author

Stefano Spezia, Jan 06 2023

Keywords

Examples

			a(3) = -16:
  [   1,  2*i, 3*i;
   -2*i,    1, 2*i;
   -3*i, -2*i,   1 ]

Links

Crossrefs

Cf. A001792 (symmetric Toeplitz matrix), A143182.
Cf. A359560 (permanent), A359561, A359562.
Cf. A359614 (minimal), A359615 (maximal).

Programs

  • Mathematica
    Join[{1},Table[Det[ToeplitzMatrix[Join[{1},I Range[2,n]]]],{n,36}]]
  • PARI
    a(n) = matdet(matrix(n, n, i, j, if (i==j, 1, if (iMichel Marcus, Jan 20 2023
  • Python
    from sympy import Matrix, I
def A359559(n): return Matrix(n,n,[i-j+(1 if i>j else -1) if i!=j else I for i in range(n) for j in range(n)]).det()*(1,-I,-1,I)[n&3] # Chai Wah Wu, Jan 25 2023

Formula

A359614(n) <= a(n) <= A359615(n).
Conjectured formulas: (Start)
O.g.f.: (1 - 5*x + 9*x^2 - 12*x^3 + 10*x^4 - 4*x^5)/(1 - 2*x + 2*x^2)^3.
a(n) = 6*a(n-1) - 18*a(n-2) + 32*a(n-3) - 36*a(n-4) + 24*a(n-5) - 8*a(n-6) for n > 5.
E.g.f.: (2 + exp(x)*((1 + x)*(2 + x)*cos(x) - (1 + x + x^2)*sin(x)))/4. (End)

A359560 a(n) is the permanent of an n X n Hermitian Toeplitz matrix whose first row consists of 1, 2*i, ..., n*i, where i denotes the imaginary unit.

Original entry on oeis.org

1, 1, 5, 18, 360, 2800, 151424, 1926704, 218991568, 3961998320, 815094714320, 19339258670304, 6524060415099520, 192715406460607360, 99364368150722162944, 3525158026102570745600, 2635328330670632415828224, 109381927750670379873854720, 113797518402277434839782802688
Offset: 0

Views

Author

Stefano Spezia, Jan 06 2023

Keywords

Examples

			a(3) = 18:
  [   1,  2*i, 3*i;
   -2*i,    1, 2*i;
   -3*i, -2*i,   1 ]

Links

Crossrefs

Cf. A143182, A204235 (symmetric Toeplitz matrix).
Cf. A359559 (determinant), A359561, A359562.
Cf. A359614 (minimal), A359615 (maximal).

Programs

  • Maple
    A359560 := proc(n)
    local T,c,r ;
    if n =0 then
        return 1 ;
    end if;
    T := Matrix(n,n,shape=hermitian) ;
    T[1,1] := 1 ;
    for c from 2 to n do
        T[1,c] := c*I ;
    end do:
    for r from 2 to n do
        for c from r to n do
            T[r,c] := T[r-1,c-1] ;
        end do:
    end do:
    LinearAlgebra[Permanent](T) ;
    simplify(%) ;
end proc:
seq(A359560(n),n=0..15) ; # R. J. Mathar, Jan 31 2023
  • Mathematica
    Join[{1},Table[Permanent[ToeplitzMatrix[Join[{1},I Range[2,n]]]],{n,18}]]
  • PARI
    a(n) = matpermanent(matrix(n, n, i, j, if (i==j, 1, if (iMichel Marcus, Jan 20 2023
  • Python
    from sympy import Matrix, I
def A359560(n): return Matrix(n,n,[i-j+(1 if i>j else -1) if i!=j else I for i in range(n) for j in range(n)]).per()*(1,-I,-1,I)[n&3] if n else 1 # Chai Wah Wu, Jan 25 2023

Formula

A359614(n) <= a(n) <= A359615(n).

A364228 Triangle read by rows: T(n, k) is the number of n X n Hermitian Toeplitz matrices of rank k using all the integers 1, 2, ..., n and with all off-diagonal elements purely imaginary.

Original entry on oeis.org

1, 0, 2, 0, 1, 5, 0, 0, 0, 24, 0, 0, 0, 0, 120, 0, 0, 0, 0, 1, 719, 0, 0, 0, 0, 0, 0, 5040, 0, 0, 0, 0, 0, 2, 1, 40317, 0, 0, 0, 0, 0, 0, 0, 6, 362874, 0, 0, 0, 0, 0, 0, 1, 0, 1, 3628798
Offset: 1

Views

Author

Stefano Spezia, Jul 14 2023

Keywords

Examples

			The triangle begins:
  1;
  0, 2;
  0, 1, 5;
  0, 0, 0, 24;
  0, 0, 0,  0, 120;
  0, 0, 0,  0,   1,  719;
  0, 0, 0,  0,   0,    0, 5040;
  0, 0, 0,  0,   0,    2,    1, 40317;
  0, 0, 0,  0,   0,    0,    0,     6, 362874;
  0, 0, 0,  0,   0,    0,    1,     0,      1, 3628798;
  ...

Links

Crossrefs

Cf. A000142 (row sums), A359614 (minimal determinant), A359615 (maximal determinant), A359616 (minimal permanent), A359617 (maximal permanent), A364229 (right diagonal).

Programs

  • Mathematica
    T[n_,k_]:=Count[Flatten[Table[MatrixRank[ToeplitzMatrix[Join[{d}, I Part[Permutations[Drop[Range[n], {d}]], i]]]],{i,(n-1)!},{d,n}]],k]; Table[T[n,k],{n,9},{k,n}]//Flatten

A364229 a(n) is the number of n X n nonsingular Hermitian Toeplitz matrices using all the integers 1, 2, ..., n and with all off-diagonal elements purely imaginary.

Original entry on oeis.org

1, 2, 5, 24, 120, 719, 5040, 40317, 362874, 3628798
Offset: 1

Views

Author

Stefano Spezia, Jul 14 2023

Keywords

Links

Crossrefs

Right diagonal of A364228.
Cf. A359614 (minimal determinant), A359615 (maximal determinant), A359616 (minimal permanent), A359617 (maximal permanent).

Programs

  • Mathematica
    a[n_]:=Count[Flatten[Table[MatrixRank[ToeplitzMatrix[Join[{d}, I Part[Permutations[Drop[Range[n], {d}]], i]]]],{i,(n-1)!},{d,n}]],n]; Array[a,9]

A359618 a(n) is the minimal absolute value of the determinant of a nonsingular n X n Hermitian Toeplitz matrix using all the integers 1, 2, ..., n and with off-diagonal elements purely imaginary.

Original entry on oeis.org

1, 1, 3, 9, 16, 21, 20, 17, 131, 62, 1
Offset: 0

Views

Author

Stefano Spezia, Jan 21 2023

Keywords

Examples

			a(4) = 16:
   [   1,   2*i,   4*i,  3*i;
    -2*i,     1,   2*i,  4*i;
    -4*i,  -2*i,     1,  2*i;
    -3*i,  -4*i,  -2*i,    1 ]

Links

Crossrefs

Cf. A348891, A358325.
Cf. A359559, A359561.
Cf. A359614 (minimal signed), A359615 (maximal signed), A359616 (minimal permanent), A359617 (maximal permanent).

Programs

  • Mathematica
    a={1}; For[n=1, n<=8, n++, mn=Infinity; For[d=1, d<=n, d++, For[i=1, i<=(n-1)!, i++, If[0<(t=Abs[Det[ToeplitzMatrix[Join[{d}, I Part[Permutations[Drop[Range[n], {d}]], i]]]]])
Showing 1-8 of 8 results.

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