OFFSET
0,2
COMMENTS
Conjecture 1. If n is a positive odd integer, x(0) = 0, and x(1),...,x(n) are variables, then the permanent of the matrix [x(j,k)]_{j,k=0..n} vanishes, where x(j,k) is 1 or (x(j)+x(k))/(x(j)-x(k)) according as j = k or not.
Conjecture 2: (-1)^n*a(n) > 0 for all n = 0,1,2,...
Conjecture 3: For any odd prime p, the permanent of the matrix [m(j,k)]_{j,k=0..p-1} is congruent to 1 - (-1)^((p-1)/2) modulo p, where m(j,k) is defined as in the title..
Conjecture 4: For any odd prime p, the permanent of the matrix M = [m(j,k)]_{j,k=1..p-1} is congruent to ((p-2)!!)^2 modulo p^2, and det(M) is congruent to ((p-2)!!)^2*(-1)^((p+1)/2)/(p-2) modulo p^2.
LINKS
Yue-Feng She, Zhi-Wei Sun, and Wei Xia, A novel permanent identity, arXiv:2208.12167 [math.CO], 2022.
Zhi-Wei Sun, New Conjectures in Number Theory and Combinatorics (in Chinese), Harbin Institute of Technology Press, 2021.
Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.
EXAMPLE
a(1) = -10 since the permanent of the matrix [m(j,k)]_{j,k=0,1,2} = [1,-1,1; 1,1,-3; 1,3,1] is -10.
MATHEMATICA
a[n_]:=a[n]=Numerator[Permanent[Table[If[j==k, 1, (j+k)/(j-k)], {j, 0, 2n}, {k, 0, 2n}]]]
Table[a[n], {n, 0, 8}]
PROG
(PARI) a(n) = numerator(matpermanent(matrix(2*n+1, 2*n+1, i, j, i--; j--; if (i==j, 1, (i+j)/(i-j))))); \\ Michel Marcus, Jul 24 2022
CROSSREFS
KEYWORD
sign
AUTHOR
Zhi-Wei Sun, Jul 24 2022
STATUS
approved