For a matrix $(a_{j,k})_{1le j,kle n}$ over a field, its permanent is defined by

$$mathrm{per}(a_{j,k})_{1le j,kle n}:=sum_{piin S_n}prod_{j=1}^n a_{j,pi(j)}.$$

In a recent preprint of mine, I investigated arithmetic properties of some permanents.

Let $n$ be a positive integer. I have proved that

$$mathrm{per}left(leftlfloorfrac{j+k}nrightrfloorright)_{1le j,kle n}=2^{n-1}+1$$

and that

$$detleft(leftlfloorfrac{j+k}nrightrfloorright)_{1le j,kle n}=(-1)^{n(n+1)/2-1}

qquadtext{if} n>1.$$

where $lfloor cdotrfloor$ is the floor function. The proofs are relatively easy.

Recall that the Bernoulli numbers $B_0,B_1,ldots$ are given by

$$frac x{e^x-1}=sum_{n=0}^infty B_nfrac{x^n}{n!} (|x|<2pi).$$

Those $G_n=2(1-2^n)B_n (n=1,2,3,ldots)$ are sometimes called Genocchi numbers.

Based on my numerical computation, here I pose the following two conjectures.

**Conjecture 1.** For any positive integer $n$, we have

$$mathrm{per}left(leftlfloorfrac{2j-k}nrightrfloorright)_{1le j,kle n}=2(2^{n+1}-1)B_{n+1}.tag{1}$$

**Conjecture 2.** For any positive integer $n$, we have

$$detleft(leftlfloorfrac{2j-k}nrightrfloorright)_{1le j,kle n}=begin{cases}(-1)^{(n^2-1)/8}&text{if} 2nmid n,\0&text{if} 2mid n.end{cases}tag{2}$$

Conjecture 2 seems easier than Conjecture 1.

**QUESTION.** Are the identities $(1)$ and $(2)$ correct? How to prove them?

Your comments are welcome!