A346949 Value of the permanent of the matrix [1-zeta^{j-k}]_{1<=j,k<=2n}, where zeta is any primitive 2n-th root of unity.

4, 48, 1440, 80640, 7257600, 958003200, 174356582400, 41845579776000, 12804747411456000, 4865804016353280000, 2248001455555215360000, 1240896803466478878720000, 806582922253211271168000000, 609776689223427721003008000000, 530505719624382117272616960000000, 526261673867387060334436024320000000

Offset: 1

Author

Zhi-Wei Sun, Aug 08 2021

Keywords

• nonn

Comments

The author has proved that the exact value of a(n) is 2*(2n)!. Moreover, for any primitive n-th root zeta of unity, the permanent of the matrix [1-zeta^j*x_k]_{1<=j,k<=n} is n!(1-x_1..x_n).

Conjecture: Let zeta be a primitive 2n-th root of unity. Then the sum of those Product_{j=1..2n}(1-zeta^{j-f(j)})^{-1} with f over all the derangements of {1,...,2n} has the exact value ((2n-1)!!/2^n)^2.

Examples

			a(1) is the permanent of the matrix [1-(-1)^{1-1},1-(-1)^{1-2};1-(-1)^{2-1},1-(-1)^{2-2}] = [0,2;2,0], which equals 4.
		

Links

• Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.

Crossrefs

Cf. A000166, A003112, A204249, A278847, A346934.

Programs

Mathematica

a[n_]:=a[n]= Permanent[Table[1-E^(2*Pi*I*(j-k)/(2*n)),{j,1,2n},{k,1,2n}]];
(* Though a(n) is actually an integer, Mathematica could not find its exact value for a general positive integer n. Instead, we may check approximate values of a(n) such as N[a[5],10] = 7257600.000. *)

PARI

default(realprecision, 100); a(n) = round(real(matpermanent(matrix(2*n, 2*n, j, k, 1-exp(Pi*I*(j-k)/n))))) \\ Michel Marcus, Aug 08 2021

Formula

a(n) = 2*(2*n)!.

Extensions

a(16) from Vaclav Kotesovec, Aug 21 2021