A359616 -id:A359616 - cp's OEIS frontend

A359614 a(n) is the minimal determinant of an n X n Hermitian Toeplitz matrix using all the integers 1, 2, ..., n and with all off-diagonal elements purely imaginary.

1, 1, -3, -30, -256, -7595, -358301, -7665804, -227965955, -13089461984, -2467071630448

Offset: 0

Stefano Spezia, Jan 07 2023

			a(4) = -256:
  [   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 ]

Wikipedia, Toeplitz Matrix.

Cf. A350953, A359559, A359561.

Cf. A359615 (maximal), A359616 (minimal permanent), A359617 (maximal permanent).

a={1}; For[n=1, n<=8, n++, mn=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]]]])

from itertools import permutations
from sympy import Matrix, I
def A359614(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)]).det()*(1,-I,-1,I)[n&3] for d in permutations(range(1,n+1))) # Chai Wah Wu, Jan 25 2023

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

Stefano Spezia, Jan 07 2023

			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 ]

Wikipedia, Toeplitz Matrix.

Cf. A350954, A359559, A359561.

Cf. A359614 (minimal), A359616 (minimal permanent), A359617 (maximal permanent).

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

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

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

Stefano Spezia, Jan 07 2023

			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 ]

Wikipedia, Toeplitz Matrix.

Cf. A351020, A359560, A359562.

Cf. A359614 (minimal determinant), A359615 (maximal determinant), A359616 (minimal).

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

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

A359561 a(n) is the determinant of an n X n Hermitian Toeplitz matrix whose first row consists of n, (n-1)i, (n-2)i, ..., 3i, 2i, i, where i denotes the imaginary unit.

Original entry on oeis.org

1, 1, 3, 0, -256, -5000, -46656, 941192, 67108864, 2066242608, 24000000000, -1659995174464, -142657607172096, -5964309791355136, -76196618232397824, 11210083593750000000, 1180591620717411303424, 62286325600853591655680, 839390038939659468275712, -213252813410122222659258368

Offset: 0

Stefano Spezia, Jan 06 2023

sign

			a(3) = 0:
  [   3,  2*i,   i;
   -2*i,    3, 2*i;
     -i, -2*i,   3 ]

Wikipedia, Toeplitz Matrix

Cf. A307783 (symmetric Toeplitz matrix).
Cf. A359559, A359560, A359562 (permanent).
Cf. A359616 (minimal), A359617 (maximal).

A359561 := proc(n)
    local T,c,r ;
    if n =0 then
        return 1 ;
    end if;
    T := Matrix(n,n) ;
    T[1,1] := n ;
    for c from 2 to n do
        T[1,c] := (n-c+1)*I ;
    end do:
    for r from 2 to n do
        for c from 1 to r-1 do
            T[r,c] := -T[c,r] ;
        end do:
        for c from r to n do
            T[r,c] := T[r-1,c-1] ;
        end do:
    end do:
    LinearAlgebra[Determinant](T) ;
    simplify(%) ;
end proc:
seq(A359561(n),n=0..25) ; # R. J. Mathar, Jan 31 2023

Join[{1},Table[Det[ToeplitzMatrix[Join[{n},I Reverse[Range[n-1]]]]],{n,19}]]

from sympy import Matrix, I
def A359561(n): return Matrix(n,n,[(n+j-i if i>j else j-i-n) if i!=j else n*I for i in range(n) for j in range(n)]).det()*(1,-I,-1,I)[n&3] # Chai Wah Wu, Jan 25 2023

A359616(n) <= a(n) <= A359617(n).

A359562 a(n) is the permanent of an n X n Hermitian Toeplitz matrix whose first row consists of n, (n-1)i, (n-2)i, ..., 3i, 2i, i, where i denotes the imaginary unit.

Original entry on oeis.org

1, 1, 5, 54, 980, 26000, 977844, 48486480, 3168454720, 257625275760, 26347709832000, 3217348801257888, 477582176242255104, 82066363639286366080, 16709994767104962690304, 3847766849105116759200000, 1029727509567022262979280896, 306114655769763238348323419392, 104188715467117934409088054935552

Offset: 0

Stefano Spezia, Jan 06 2023

nonn

			a(3) = 54:
  [   3,  2*i,   i;
   -2*i,    3, 2*i;
     -i, -2*i,   3 ]

Wikipedia, Toeplitz Matrix

Cf. A307783 (symmetric Toeplitz matrix).
Cf. A359559, A359560, A359561 (determinant).
Cf. A359616 (minimal), A359617 (maximal).

Join[{1},Table[Permanent[ToeplitzMatrix[Join[{n},I Reverse[Range[n-1]]]]],{n,18}]]

from sympy import Matrix, I
def A359562(n): return Matrix(n,n,[(n+j-i if i>j else j-i-n) if i!=j else n*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

A359616(n) <= a(n) <= A359617(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

Stefano Spezia, Jul 14 2023

			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;
  ...

Wikipedia, Toeplitz Matrix

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

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

Stefano Spezia, Jul 14 2023

Wikipedia, Toeplitz Matrix

Right diagonal of A364228.

Cf. A359614 (minimal determinant), A359615 (maximal determinant), A359616 (minimal permanent), A359617 (maximal permanent).

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

Stefano Spezia, Jan 21 2023

			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 ]

Wikipedia, Toeplitz Matrix

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

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]]]]])

cp's OEIS Frontend

A359614 a(n) is the minimal determinant of an n X n Hermitian Toeplitz matrix using all the integers 1, 2, ..., n and with all off-diagonal elements purely imaginary.

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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.

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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.

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A359561 a(n) is the determinant of an n X n Hermitian Toeplitz matrix whose first row consists of n, (n-1)*i, (n-2)*i, ..., 3*i, 2*i, i, where i denotes the imaginary unit.

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A359562 a(n) is the permanent of an n X n Hermitian Toeplitz matrix whose first row consists of n, (n-1)*i, (n-2)*i, ..., 3*i, 2*i, i, where i denotes the imaginary unit.

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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.

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Mathematica

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.

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Mathematica

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.

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Author

Keywords

Examples

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Mathematica

A359561 a(n) is the determinant of an n X n Hermitian Toeplitz matrix whose first row consists of n, (n-1)i, (n-2)i, ..., 3i, 2i, i, where i denotes the imaginary unit.

A359562 a(n) is the permanent of an n X n Hermitian Toeplitz matrix whose first row consists of n, (n-1)i, (n-2)i, ..., 3i, 2i, i, where i denotes the imaginary unit.