Permanent identities, combinatorial sequences, and permutation statistics
Shishuo Fu, Zhicong Lin, Zhi-Wei Sun
Abstract
In this paper, we confirm six conjectures on the exact values of some permanents, relating them to the Genocchi numbers of the first and second kinds as well as the Euler numbers. For example, we prove that $$\mathrm{per}\left[\left\lfloor\frac{2j-k}{n}\right\rfloor\right]_{1\le j,k\le n}=2(2^{n+1}-1)B_{n+1},$$ where $B_0,B_1,B_2,\ldots$ are the Bernoulli numbers. We also show that $$ \mathrm{per}\left[\mathrm{sgn}\left(\cosπ\frac{i+j}{n+1}\right)\right]_{1\le i,j\le n}=\begin{cases} -\sum_{k=0}^m\binom{m}{k}E_{2k+1}&\quad\text{if}\ n=2m+1,\\ \sum_{k=0}^m\binom{m}{k}E_{2k}&\quad\text{if}\ n=2m, \end{cases} $$ where $\mathrm{sgn}(x)$ is the sign function, and $E_0,E_1,E_2,\ldots$ are the Euler (zigzag) numbers. In the course of linking the evaluation of these permanents to the aforementioned combinatorial sequences, the classical permutation statistic -- the excedance number, together with several kinds of its variants, plays a central role. Our approach features recurrence relations, bijections, as well as certain elementary operations on matrices that preserve their permanents. Moreover, our proof of the second permanent identity leads to a proof of Bala's conjectural continued fraction formula, and an unexpected permutation interpretation for the $γ$-coefficients of the $2$-Eulerian polynomials.