Coxeter Groups and Unimodality — Conjecture 8

Definitions and mathematical statement

For a positive integer $n$, let $[n]=\{1,\ldots,n\}$, and let $S_n$ be the set of all permutations of $[n]$. For $\sigma\in S_n$, define its odd length, also called its odd inversion number, by \[ L(\sigma)=\left|\left\{(i,j)\in[n]^2:i<j,\ \sigma(i)>\sigma(j),\ i\not\equiv j\pmod 2\right\}\right|. \] Define \[ L_n(x)=\sum_{\sigma\in S_n}x^{L(\sigma)} =\sum_{k=0}^{d_n}a_{n,k}x^k, \qquad d_n=\deg L_n(x), \] where $a_{n,k}=|\{\sigma\in S_n:L(\sigma)=k\}|$.

The supplied conjecture states that, for every $n\in\mathbb{P}$ with $n\geq5$, the polynomial $L_n(x)$ is symmetric and unimodal. Explicitly, the required properties are \[ a_{n,k}=a_{n,d_n-k}\qquad(0\leq k\leq d_n) \] and the existence of an integer $m\in\{0,\ldots,d_n\}$ such that \[ a_{n,0}\leq\cdots\leq a_{n,m}\geq\cdots\geq a_{n,d_n}. \] The supplied record does not define $\mathbb{P}$. Its notation is retained here: whether the quantifier ranges over all positive integers or only primes requires clarification from the source. The record labels this assertion open; that status is supplied metadata, not an independently verified mathematical assessment.

Short English statement

Count inversions whose two positions have opposite parity. The conjecture asserts that, for each permitted $n\geq5$, the numbers of permutations having each possible odd inversion number form a symmetric, unimodal sequence. The source's domain notation $\mathbb{P}$ remains unresolved.

Sources

• Local UnsolvedMath registry, permanent ID $800370$, exact source problem code catalogue entry, titled "Coxeter Groups and Unimodality --- Conjecture 8." The record attributes the problem to Brenti (2024) and supplies the status open.
• Brenti, Some Open Problems on Coxeter Groups and Unimodality (2024), Conjecture 8, https://arxiv.org/abs/2410.09897. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent source access.
• The supplied statement identifies the conjecture as [BreCa2, Conj. 6.2]. It also reports that the statistic was introduced in [KloVol] in relation to formed spaces and was further studied in [BreCav] and [BreCa2]. Complete bibliographic details for these reference labels are not supplied.