A359560 -id:A359560 - cp's OEIS frontend

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.

1, 1, 5, 18, 245, 2249, 57213, 947177, 50431724, 1282453618

Offset: 0

Stefano Spezia, Jan 07 2023

			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 ]

Wikipedia, Toeplitz Matrix.

Cf. A351019, A359560, A359562.

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

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]]]])

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

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

A359559 a(n) is the determinant of an n X n Hermitian Toeplitz matrix whose first row consists of 1, 2i, ..., ni, 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

Stefano Spezia, Jan 06 2023

sign

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

Wikipedia, Toeplitz Matrix

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

Join[{1},Table[Det[ToeplitzMatrix[Join[{1},I Range[2,n]]]],{n,36}]]

a(n) = matdet(matrix(n, n, i, j, if (i==j, 1, if (iMichel Marcus, Jan 20 2023

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

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)

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).

cp's OEIS Frontend

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.

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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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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.

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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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Author

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Programs

Mathematica

Python

Formula

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

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.