A306714 -id:A306714 - cp's OEIS frontend

A008305 Triangle read by rows: a(n,k) = number of permutations of [n] allowing i->i+j (mod n), j=0..k-1.

1, 1, 2, 1, 2, 6, 1, 2, 9, 24, 1, 2, 13, 44, 120, 1, 2, 20, 80, 265, 720, 1, 2, 31, 144, 579, 1854, 5040, 1, 2, 49, 264, 1265, 4738, 14833, 40320, 1, 2, 78, 484, 2783, 12072, 43387, 133496, 362880, 1, 2, 125, 888, 6208, 30818, 126565, 439792, 1334961, 3628800

Offset: 1

N. J. A. Sloane

The point is, we are counting permutations of [n] = {1,2,...,n} with the restriction that i cannot move by more than k places. Hence the phrase "permutations with restricted displacements". - N. J. A. Sloane, Mar 07 2014

The triangle could have been defined as an infinite square array by setting a(n,k) = n! for k >= n.

			a(4,3) = 9 because 9 permutations of {1,2,3,4} are allowed if each i can be placed on 3 positions i+0, i+1, i+2 (mod 4): 1234, 1423, 1432, 3124, 3214, 3412, 4123, 4132, 4213.
Triangle begins:
  1,
  1, 2,
  1, 2,   6,
  1, 2,   9,  24,
  1, 2,  13,  44,  120,
  1, 2,  20,  80,  265,   720,
  1, 2,  31, 144,  579,  1854,   5040,
  1, 2,  49, 264, 1265,  4738,  14833,  40320,
  1, 2,  78, 484, 2783, 12072,  43387, 133496,  362880,
  1, 2, 125, 888, 6208, 30818, 126565, 439792, 1334961, 3628800,
  ...

H. Minc, Permanents, Encyc. Math. #6, 1978, p. 48

Alois P. Heinz, Rows n = 1..23, flattened
Henry Beker and Chris Mitchell, Permutations with restricted displacement, SIAM J. Algebraic Discrete Methods 8 (1987), no. 3, 338--363. MR0897734 (89f:05009)
N. S. Mendelsohn, Permutations with confined displacement, Canad. Math. Bull., 4 (1961), 29-38.
N. Metropolis, M. L. Stein, P. R. Stein, Permanents of cyclic (0,1) matrices, J. Combin. Theory, 7 (1969), 291-321.
Wikipedia, Permanent (mathematics)

Diagonals (from the right): A000142, A000166, A000179, A000183, A004307, A189389, A184965.
Diagonals (from the left): A000211 or A048162, 4*A000382 or A004306 or A000803, A000804, A000805.
a(n,ceiling(n/2)) gives A306738.
Cf. A002467, A306714.

with(LinearAlgebra):
a:= (n, k)-> Permanent(Matrix(n,
             (i, j)-> `if`(0<=j-i and j-i

a[n_, k_] := Permanent[Table[If[0 <= j-i && j-i < k || j-i < k-n, 1, 0], {i, 1,n}, {j, 1, n}]]; Table[Table[a[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)

a(n,k) = per(sum(P^j, j=0..k-1)), where P is n X n, P[ i, i+1 (mod n) ]=1, 0's otherwise.

a(n,n) - a(n,n-1) = A002467(n). - Alois P. Heinz, Mar 06 2019

Comments and more terms from Len Smiley
More terms from Vladeta Jovovic, Oct 02 2003
Edited by Alois P. Heinz, Dec 18 2010

A306595 Determinant of the circulant matrix whose first column corresponds to the binary digits of n.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 2, 0, 1, 0, 0, 3, 0, -3, 3, 0, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 0, 1, 0, 4, 0, 0, -9, 9, 0, 4, 9, 0, 8, 9, 0, 8, 5, 0, 0, 9, 0, -9, -8, 0, -5, 0, 0, 8, 5, 0, -5, 5, 0, 1, 2, 2, 3, 2, 24, 24, 4, 2, 3, 3, 32, 3, 4, 32, 5, 2, 24, 3

Offset: 0

Rémy Sigrist, Feb 27 2019

This sequence is the binary variant of A177894.
From Robert Israel, Mar 05 2019: (Start)
a(n) is divisible by A000120(n).
If A070939(n) is even then n is divisible by A000120(n)*A065359(n). (End)

			For n = 13:
- the binary representation of 13 is "1101",
- the corresponding circulant matrix is:
    [1 1 0 1]
    [1 1 1 0]
    [0 1 1 1]
    [1 0 1 1]
- its determinant is -3,
- hence a(13) = -3.

Robert Israel, Table of n, a(n) for n = 0..10000
Wikipedia, Circulant matrix
Index entries for sequences related to binary expansion of n

Cf. A000120, A065359, A070939, A121016, A177894, A219325, A306714.

a:= n-> `if`(n=1, 1, (l-> LinearAlgebra[Determinant](Matrix(nops(l),
       shape=Circulant[l[-i]$i=1..nops(l)])))(convert(n, base, 2))):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 05 2019

a(n) = my (d=if (n, binary(n), [0])); my (m=matrix(#d, #d, i,j, d[1+(i-j)%#d])); return (matdet(m))

a(A121016(n)) = 0 for any n > 0.
a(2^k) = 1 for any k >= 0.
a(A219325(n)) = A219325(n) for any n > 0.

A306853 Positive integers equal to the permanent of the circulant matrix formed by their decimal digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 261, 370, 407, 52036, 724212, 223123410

Offset: 1

Paolo P. Lava, Mar 13 2019

1, 2, 3, 4, 5, 6, 7, 8, 9, 370 and 407 are also equal to the determinant of the circulant matrix formed by their decimal digits.

			     | 2 6 1 |
perm | 1 2 6 | = 2*2*2 + 6*6*6 + 1*1*1 + 1*2*6 + 6*1*2 + 2*6*1 = 261.
     | 6 1 2 |
.
     | 2 2 3 1 2 3 4 1 0 |
     | 0 2 2 3 1 2 3 4 1 |
     | 1 0 2 2 3 1 2 3 4 |
     | 4 1 0 2 2 3 1 2 3 |
perm | 3 4 1 0 2 2 3 1 2 | = 223123410
     | 2 3 4 1 0 2 2 3 1 |
     | 1 2 3 4 1 0 2 2 3 |
     | 3 1 2 3 4 1 0 2 2 |
     | 2 3 1 2 3 4 1 0 2 |

Eric Weisstein's World of Mathematics, Permanent
Eric Weisstein's World of Mathematics, Circulant Matrix

Cf. A219324, A219327, A306662, A306593, A306714.

Up to n=110 the permanent of the circulant matrix of the digits of n is equal to A101337 but from n=111 on it can differ.

with(linalg): P:=proc(q) local a, b, c, d, i, j, k, n, t;
for n from 1 to q do d:=ilog10(n)+1; a:=convert(n, base, 10); c:=[];
for k from 1 to nops(a) do c:=[op(c), a[-k]]; od; t:=[op([]), c];
for k from 2 to d do b:=[op([]), c[nops(c)]];
for j from 1 to nops(c)-1 do b:=[op(b), c[j]]; od;
c:=b; t:=[op(t), c]; od; if n=permanent(t)
then print(n); fi; od; end: P(10^7);

mpd(n) = {my(d = digits(n)); matpermanent(matrix(#d, #d, i, j, d[1+lift(Mod(j-i, #d))]));}
isok(n) = mpd(n) == n; \\ Michel Marcus, Mar 14 2019

from sympy import Matrix
A306853_list = []
for n in range(1,10**6):
    s = [int(d) for d in str(n)]
    m = len(s)
    if n == Matrix(m, m, lambda i, j: s[(i-j) % m]).per():
        A306853_list.append(n) # Chai Wah Wu, Oct 18 2021

a(15) from Vaclav Kotesovec, Aug 19 2021

A308126 Positive integers equal to the permanent of Hankel matrix formed by their decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 562, 962, 26240, 85440

Offset: 1

Paolo P. Lava, May 14 2019

			     | 5 6 2 |
perm | 6 2 6 | = 5*2*5 + 6*6*2 + 2*6*6 + 2*2*2 + 6*6*5 + 5*6*6 = 562.
     | 2 6 5 |

Eric Weisstein's World of Mathematics, Permanent
Wikipedia, Hankel matrix

Cf. A306714, A306853.

with(linalg): P:=proc(q) local c, d, k, n, t: print(0);
for n from 1 to q do c:=convert(n, base, 10): t:=[]:
for k from 1 to nops(c) do t:=[op(t), 0]: od: d:=t: t:=[]:
for k from 1 to nops(c) do t:=[op(t), d]: t[k, -k]:=1: od:
if permanent(evalm(toeplitz(c) &* t))=n then print(n); fi:
od: end: P(10^8);

cp's OEIS Frontend

A008305 Triangle read by rows: a(n,k) = number of permutations of [n] allowing i->i+j (mod n), j=0..k-1.

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A306595 Determinant of the circulant matrix whose first column corresponds to the binary digits of n.

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A306853 Positive integers equal to the permanent of the circulant matrix formed by their decimal digits.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Maple

PARI

Python

Extensions

A308126 Positive integers equal to the permanent of Hankel matrix formed by their decimal digits.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple