A322908 -id:A322908 - cp's OEIS frontend

A350931 Maximal determinant of an n X n Toeplitz matrix using the integers 1 to 2*n - 1.

1, 1, 7, 105, 2294, 71753, 3051554, 175457984

Offset: 0

Stefano Spezia, Jan 25 2022

Also maximal absolute value of the determinant of an n X n Hankel matrix using the integers 1 to 2*n - 1. - Stefano Spezia, Dec 22 2023

			a(2) = 7:
   [3    1]
   [2    3]
a(3) = 105:
   [5    1    3]
   [4    5    1]
   [2    4    5]

Lucas A. Brown, A350930+1.py
Wikipedia, Toeplitz Matrix

Cf. A322908, A323254, A350930 (minimal).

from itertools import permutations
from sympy import Matrix
def A350931(n): return max(Matrix([p[n-1-i:2*n-1-i] for i in range(n)]).det() for p in permutations(range(1,2*n))) # Chai Wah Wu, Jan 27 2022

a(5) from Alois P. Heinz, Jan 25 2022
a(6) from Pontus von Brömssen, Jan 26 2022
a(7) from Lucas A. Brown, Aug 28 2022

A350937 Minimal permanent of an n X n Toeplitz matrix using the integers 1 to 2*n - 1.

Original entry on oeis.org

1, 1, 7, 89, 2287, 89025, 5141775, 404316249

Offset: 0

Stefano Spezia, Jan 26 2022

At least up to a(7) the minimal permanent is attained by a matrix which has 1, 3, 5, ... as first row and 1, 2, 4, 6,... as first column. - Giovanni Resta, Oct 13 2022

Also minimal permanent of an n X n Hankel matrix using the integers 1 to 2*n - 1. - Stefano Spezia, Dec 22 2023

			a(2) = 7:
    1    2
    3    1
a(3) = 89:
    1    2    4
    3    1    2
    5    3    1

Lucas A. Brown, A350937+8.sage
Wikipedia, Toeplitz Matrix

Cf. A322908, A323254, A350930, A350938 (maximal).

from itertools import permutations
from sympy import Matrix
def A350937(n): return 1 if n == 0 else min(Matrix([p[n-1-i:2*n-1-i] for i in range(n)]).per() for p in permutations(range(1,2*n))) # Chai Wah Wu, Jan 27 2022

a(5) from Alois P. Heinz, Jan 26 2022
a(6) from Lucas A. Brown, Sep 04 2022
a(7) from Giovanni Resta, Oct 13 2022

A350938 Maximal permanent of an n X n Toeplitz matrix using the integers 1 to 2*n - 1.

Original entry on oeis.org

1, 1, 11, 296, 14502, 1153889, 134713213, 21788125930

Offset: 0

Stefano Spezia, Jan 26 2022

Also maximal permanent of an n X n Hankel matrix using the integers 1 to 2*n - 1. - Stefano Spezia, Dec 22 2023

			a(2) = 11:
    3    1
    2    3
a(3) = 296:
    5    3    2
    4    5    3
    1    4    5

Lucas A. Brown, A350937+8.sage
Wikipedia, Toeplitz Matrix

Cf. A322908, A323254, A350931, A350937 (minimal).

from itertools import permutations
from sympy import Matrix
def A350938(n): return 1 if n == 0 else max(Matrix([p[n-1-i:2*n-1-i] for i in range(n)]).per() for p in permutations(range(1,2*n))) # Chai Wah Wu, Jan 27 2022

a(5) from Alois P. Heinz, Jan 26 2022
a(6) from Lucas A. Brown, Sep 04 2022
a(7) from Giovanni Resta, Oct 13 2022

A323254 The determinant of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 2n - 1, n - 1, ..., 1 and whose first column consists of 2n - 1, 2*n - 2, ..., n.

Original entry on oeis.org

1, 7, 58, 614, 8032, 125757, 2298208, 48075148, 1133554432, 29756555315, 860884417024, 27218972906226, 933850899349504, 34556209025624041, 1371957513591119872, 58174957356247084568, 2624017129323317493760, 125454378698728779884895, 6337442836338834419089408

Offset: 1

Stefano Spezia, Jan 09 2019

nonn

The trace of the matrix M(n) is A000384(n). [Corrected by Stefano Spezia, Dec 08 2019]
The sum of the first row of the matrix M(n) is A034856(n).
The sum of the first column of the matrix M(n) is A000326(n).
For n > 1, the sum of the superdiagonal of the matrix M(n) is A000290(n-1).
For n > 1, the sum of the subdiagonal of the matrix M(n) is A001105(n-1).

			For n = 1 the matrix M(1) is
   1
with determinant Det(M(1)) = 1.
For n = 2 the matrix M(2) is
   3, 1
   2, 3
with Det(M(2)) = 7.
For n = 3 the matrix M(3) is
   5, 2, 1
   4, 5, 2
   3, 4, 5
with Det(M(3)) = 58.

Vaclav Kotesovec, Table of n, a(n) for n = 1..300
Wikipedia, Toeplitz matrix

Cf. A000290, A000326, A000384, A001105, A001792.
Cf. A034856, A204235, A318173, A322908, A322909.
Cf. A323255 (permanent of matrix M(n)).

b[i_]:=i; a[n_]:=Det[ToeplitzMatrix[Join[{b[2*n-1]}, Array[b, n-1, {2*n-2,n}]], Join[{b[2*n-1]},Array[b, n-1, {n-1,1}]]]]; Array[a,20]

tm(n) = {my(m = matrix(n, n, i, j, if (j==1, 2*n-i, n-j+1))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m;}
a(n) = matdet(tm(n)); \\ Stefano Spezia, Dec 11 2019

a(n) ~ (5*exp(1) + exp(-1)) * n^n / 4. - Vaclav Kotesovec, Jan 10 2019

A350930 Minimal determinant of an n X n Toeplitz matrix using the integers 1 to 2*n - 1.

Original entry on oeis.org

1, 1, -5, -42, -1810, -48098, -2737409, -114381074

Offset: 0

Stefano Spezia, Jan 25 2022

			a(2) = -5:
   [1    2]
   [3    1]
a(3) = -42:
   [3    4    1]
   [5    3    4]
   [2    5    3]

Lucas A. Brown, A350930+1.py
Wikipedia, Toeplitz Matrix

Cf. A322908, A323254, A350931 (maximal).

from itertools import permutations
from sympy import Matrix
def A350930(n): return min(Matrix([p[n-1-i:2*n-1-i] for i in range(n)]).det() for p in permutations(range(1,2*n))) # Chai Wah Wu, Jan 27 2022

a(5) from Alois P. Heinz, Jan 25 2022
a(6) from Pontus von Brömssen, Jan 26 2022
a(7) from Lucas A. Brown, Aug 28 2022

A322909 The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 1, ..., n and whose first column consists of 1, n + 1, ..., 2*n - 1.

Original entry on oeis.org

1, 1, 7, 100, 2840, 129428, 8613997, 791557152, 95921167710, 14818153059968, 2842735387366627, 663020104070865664, 184757202542187563476, 60623405966739216871680, 23135486197103263598936745, 10160292704659539620791062528, 5087671168376607498331875818106

Offset: 0

Stefano Spezia, Dec 30 2018

nonn

The matrix M(n) differs from that of A306457 in using successive positive integers in place of successive prime numbers. [Modified by Stefano Spezia, Dec 20 2019 at the suggestion of Michel Marcus]
The trace of M(n) is A000027(n).
The sum of the first row of M(n) is A000217(n).
The sum of the first column of M(n) is A005448(n). [Corrected by Stefano Spezia, Dec 19 2019]
For n > 1, the sum of the superdiagonal of M(n) is A005843(n).
For n > 0, the sum of the (k-1)-th superdiagonal of M(n) is A003991(n,k). - Stefano Spezia, Dec 29 2019
For n > 1 and k > 0, the sum of the k-th subdiagonal of M(n) is A120070(n,k). - Stefano Spezia, Dec 31 2019

			For n = 1 the matrix M(1) is
   1
with permanent a(1) = 1.
For n = 2 the matrix M(2) is
   1, 2
   3, 1
with permanent a(2) = 7.
For n = 3 the matrix M(3) is
   1, 2, 3
   4, 1, 2
   5, 4, 1
with permanent a(3) = 100.

Stefano Spezia, Table of n, a(n) for n = 0..35
Wikipedia, Toeplitz Matrix

Cf. A000027, A000217, A003991, A005448, A005843, A120070, A306457, A322908 (determinant of M(n)).

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(ToeplitzMatrix([
         seq(i, i=2*n-1..n+1, -1), seq(i, i=1..n)]))):
seq(a(n), n = 0 .. 15);

b[n_]:=n; a[n_]:=If[n==0,1,Permanent[ToeplitzMatrix[Join[{b[1]}, Array[b, n-1, {n+1, 2*n-1}]], Array[b, n]]]]; Array[a, 15,0]

tm(n) = {my(m = matrix(n, n, i, j, if (i==1, j, if (j==1, n+i-1)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
a(n) = matpermanent(tm(n)); \\ Stefano Spezia, Dec 19 2019

a(0) = 1 prepended by Stefano Spezia, Dec 19 2019

A323255 The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 2n - 1, n - 1, ..., 1 and whose first column consists of 2n - 1, 2*n - 2, ..., n.

Original entry on oeis.org

1, 1, 11, 248, 9968, 638772, 60061657, 7798036000, 1336715859150, 292406145227392, 79483340339739367, 26280500564448081664, 10386012861097225139356, 4834639222955142417477888, 2618110215141486526589786501, 1631888040186649673361825042432, 1159983453675106278249250918734938

Offset: 0

Stefano Spezia, Jan 09 2019

nonn

The trace of the matrix M(n) is A000384(n).
The sum of the first row of the matrix M(n) is A034856(n).
The sum of the first column of the matrix M(n) is A000326(n).
For n > 1, the sum of the superdiagonal of the matrix M(n) is A000290(n-1).
For n > 1, the sum of the subdiagonal of the matrix M(n) is A001105(n-1).

			For n = 1 the matrix M(1) is
   1
with permanent a(1) = 1.
For n = 2 the matrix M(2) is
   3, 1
   2, 3
with permanent a(2) = 11.
For n = 3 the matrix M(3) is
   5, 2, 1
   4, 5, 2
   3, 4, 5
with permanent a(3) = 248.

Stefano Spezia, Table of n, a(n) for n = 0..35
Wikipedia, Toeplitz matrix

Cf. A000290, A000326, A000384, A001105, A001792.
Cf. A034856, A204235, A318173, A322908, A322909.
Cf. A323254 (determinant of matrix M(n)).

b[i_]:=i; a[n_]:=If[n==0, 1, Permanent[ToeplitzMatrix[Join[{b[2*n-1]}, Array[b, n-1, {2*n-2,n }]], Join[{b[2*n-1]},Array[b, n-1, {n-1,1}]]]]]; Array[a, 16, 0]

tm(n) = {my(m = matrix(n, n, i, j, if (j==1, 2*n-i, n-j+1))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m;}
a(n) = matpermanent(tm(n)); \\ Stefano Spezia, Dec 11 2019

a(0) = 1 prepended by Stefano Spezia, Dec 08 2019

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A350931 Maximal determinant of an n X n Toeplitz matrix using the integers 1 to 2*n - 1.

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A350937 Minimal permanent of an n X n Toeplitz matrix using the integers 1 to 2*n - 1.

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A350938 Maximal permanent of an n X n Toeplitz matrix using the integers 1 to 2*n - 1.

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A323254 The determinant of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 2*n - 1, n - 1, ..., 1 and whose first column consists of 2*n - 1, 2*n - 2, ..., n.

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A350930 Minimal determinant of an n X n Toeplitz matrix using the integers 1 to 2*n - 1.

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A322909 The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 1, ..., n and whose first column consists of 1, n + 1, ..., 2*n - 1.

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A323255 The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 2*n - 1, n - 1, ..., 1 and whose first column consists of 2*n - 1, 2*n - 2, ..., n.

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A323254 The determinant of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 2n - 1, n - 1, ..., 1 and whose first column consists of 2n - 1, 2*n - 2, ..., n.

A323255 The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 2n - 1, n - 1, ..., 1 and whose first column consists of 2n - 1, 2*n - 2, ..., n.