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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.

%I A322277 #13 Dec 21 2018 03:27:28
%S A322277 1,11,490,60916,15745548,7477647372,5799397213200,6925325038489152,
%T A322277 11958227405868674880,28853103567727115409600,
%U A322277 93561657023119005869616000,398720531811315564754326938880,2174628314166392755825875267321600,14941853448103858870808931238617312000
%N 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.
%C A322277 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.
%C A322277 det(M(1)) = 1, det(M(2)) = -5 and det(M(n)) = 0 for n > 2 (proved).
%C A322277 The trace of the matrix M(n) is A317614(n).
%H A322277 Vaclav Kotesovec, <a href="/A322277/b322277.txt">Table of n, a(n) for n = 1..35</a>
%e A322277 For n = 1 the matrix M(1) is
%e A322277   1
%e A322277 with permanent a(1) = 1.
%e A322277 For n = 2 the matrix M(2) is
%e A322277   1, 2
%e A322277   4, 3
%e A322277 with permanent a(2) = 11.
%e A322277 For n = 3 the matrix M(3) is
%e A322277   1, 2, 3
%e A322277   6, 5, 4
%e A322277   7, 8, 9
%e A322277 with permanent a(3) = 490.
%p A322277 with(LinearAlgebra):
%p A322277 a := n -> Permanent(Matrix(n, (i, j) -> 1-j+i*n+(-1+2*j-n)*modp(i,2))):
%p A322277 seq(a(n), n = 1 .. 20);
%t A322277 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]
%o A322277 (PARI)
%o A322277 a(n) = matpermanent(matrix(n, n, i, j, if (i % 2, j + n*(i-1), n*i - j + 1)));
%o A322277 vector(20, n, a(n))
%Y A322277 Cf. A317614 (trace of matrix M(n)).
%Y A322277 Cf. A241016 (row sums of M matrices), A317617 (column sums of M matrices), A074147 (antidiagonals of M matrices).
%K A322277 nonn
%O A322277 1,2
%A A322277 _Stefano Spezia_, Dec 01 2018