# nt.number theory – A novel identity connecting permanents to Bernoulli numbers

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?