

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 zigzag pattern.


11



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

1,2


COMMENTS

M(n) is defined as M[i,j,n] = j + n*(i1) 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).


LINKS



EXAMPLE

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.


MAPLE

with(LinearAlgebra):
a := n > Permanent(Matrix(n, (i, j) > 1j+i*n+(1+2*jn)*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]


PROG

(PARI)
a(n) = matpermanent(matrix(n, n, i, j, if (i % 2, j + n*(i1), n*i  j + 1)));
vector(20, n, a(n))


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).


KEYWORD

nonn


AUTHOR



STATUS

approved



