A322277 Permanent of an n X n square matrix M(n) formed by writing the numbers 1, ..., n^2 successively forward and backward along the rows in zig-zag pattern.

1, 11, 490, 60916, 15745548, 7477647372, 5799397213200, 6925325038489152, 11958227405868674880, 28853103567727115409600, 93561657023119005869616000, 398720531811315564754326938880, 2174628314166392755825875267321600, 14941853448103858870808931238617312000

Offset: 1

Author

Stefano Spezia, Dec 01 2018

Keywords

• nonn

Comments

M(n) is defined as M[i,j,n] = j + n*(i-1) if i is odd and M[i,j,n] = n*i - j + 1 if i is even.

det(M(1)) = 1, det(M(2)) = -5 and det(M(n)) = 0 for n > 2 (proved).

The trace of the matrix M(n) is A317614(n).

Examples

			For n = 1 the matrix M(1) is
  1
with permanent a(1) = 1.
For n = 2 the matrix M(2) is
  1, 2
  4, 3
with permanent a(2) = 11.
For n = 3 the matrix M(3) is
  1, 2, 3
  6, 5, 4
  7, 8, 9
with permanent a(3) = 490.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..35

Crossrefs

Cf. A317614 (trace of matrix M(n)).

Cf. A241016 (row sums of M matrices), A317617 (column sums of M matrices), A074147 (antidiagonals of M matrices).

Programs

Maple

with(LinearAlgebra):
a := n -> Permanent(Matrix(n, (i, j) -> 1-j+i*n+(-1+2*j-n)*modp(i,2))):
seq(a(n), n = 1 .. 20);

Mathematica

M[i_, j_, n_] := 1 - j + i n + (-1 + 2 j - n) Mod[i, 2]; a[n_] := Permanent[Table[M[i, j, n], {i, n}, {j, n}]]; Array[a, 20]

PARI

a(n) = matpermanent(matrix(n, n, i, j, if (i % 2, j + n*(i-1), n*i - j + 1)));
vector(20, n, a(n))