For a matrix $[a_{j,k}]_{1\le j,k\le n}$ over a field, its permanent is defined by $$\mathrm{per}[a_{j,k}]_{1\le j,k\le n}:=\sum_{\pi\in S_n}\prod_{j=1}^n a_{j,\pi(j)}.$$ In a recent preprint of mine, I investigated arithmetic properties of some permanents.

Let $n$ be a positive integer. I have proved that $$\mathrm{per}\left[\left\lfloor\frac{j+k}n\right\rfloor\right]_{1\le j,k\le n}=2^{n-1}+1$$ and that $$\det\left[\left\lfloor\frac{j+k}n\right\rfloor\right]_{1\le j,k\le n}=(-1)^{n(n+1)/2-1} \qquad\text{if}\ \ n>1.$$ where $\lfloor \cdot\rfloor$ is the floor function. The proofs are relatively easy.

Recall that the Bernoulli numbers $B_0,B_1,\ldots$ are given by $$\frac x{e^x-1}=\sum_{n=0}^\infty B_n\frac{x^n}{n!}\ \ \ \ (|x|<2\pi).$$ 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
$$\mathrm{per}\left[\left\lfloor\frac{2j-k}n\right\rfloor\right]_{1\le j,k\le n}=2(2^{n+1}-1)B_{n+1}.\tag{1}$$

**Conjecture 2.** For any positive integer $n$, we have
$$\det\left[\left\lfloor\frac{2j-k}n\right\rfloor\right]_{1\le j,k\le n}=\begin{cases}(-1)^{(n^2-1)/8}&\text{if}\ 2\nmid n,\\0&\text{if}\ 2\mid n.\end{cases}\tag{2}$$

Conjecture 2 seems easier than Conjecture 1.

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

Your comments are welcome!

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