Permanent identities, combinatorial sequences, and permutation statistics

TL;DR

This paper proves six conjectures relating permanents to classical combinatorial sequences like Bernoulli, Euler, and Genocchi numbers, using permutation statistics and matrix operations, and connects these results to continued fractions and polynomial coefficients.

Contribution

It confirms six conjectures on permanents linked to well-known combinatorial numbers and introduces new permutation interpretations and proof techniques involving recurrence relations and matrix operations.

Findings

Confirmed six conjectures on permanents and combinatorial sequences.

Established new permutation interpretations for Eulerian polynomial coefficients.

Linked permanent evaluations to Bala's continued fraction conjecture.

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 $$ \mathrm{per}\left[\mathrm{sgn}\left(\cos\pi\frac{i+j}{n+1}\right)\right]_{1\le i,j\le n}=\begin{cases} -\sum_{k=0}^m\binom{m}{k}E_{2k+1}&\quad\text{if}\ n=2m+1,\\ \sum_{k=0}^m\binom{m}{k}E_{2k}&\quad\text{if}\ n=2m, \end{cases} $$ 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 $\gamma$-coefficients of the $2$-Eulerian polynomials.