Noncrossing partitions, noncrossing graphs, and \(q\)-permanental equations
Given an \(n\times n\) matrix \(A=(a_{ij})\) and a real number \(\mu\), we define the \(\mu\)-permanent of \(A\) as the polynomial \[ P_\mu(A)=\sum_{\sigma\in S_n} a_{1\sigma(1)}\cdots a_{n\sigma(n)} \mu^{\ell(\sigma)}\; , \] where \(\ell(\sigma)\) is the number of inversions of the permutation \(\sigma\) in the symmetric group \(S_n\) of degree \(n\), i.e., \(\ell(\sigma)=\#\{(i,j)\in \{1,\ldots,n\}^2\, |\, i< j \; \text{ and }\; \sigma(i)>\sigma(j)\}\). In [Linear Multilinear Algebra 32, 335--337 (1992; Zbl 0765.15008)], \textit{R. B. Bapat} conjectured that \(P_\mu(A)\) was a strictly increasing function of \(\mu\in [-1,1]\), proving it for \(n=2,3\) and for tridiagonal matrices. Later on, the reviewer [Linear Multilinear Algebra 53, 225--230 (2005; Zbl 1078.15007)] extended this result for a particular kind of acyclic matrices. In \(2016\) it was presented by \textit{M. Anđelić}, \textit{A. Pereira}, and the reviewer in [``The \(\mu\)-permanent, a new graph labeling, and a known integer sequence'', Preprint, \url{arXiv:1609.04208}] a general condition on the labeling of the trees for such property to be satisfied. Namely, given two disjoint edges \(ij\) and \(k\ell\) (say \(i< j\), \(k< \ell\), and \(i< k\)), then one of the conditions is satisfied: (1) \(i< j< k< \ell\); (2) \(i< k< \ell< j\). Less known references were also presented there. In the paper under review, the author recasts these results in terms of noncrossing partitions and the \(\mu\)-permanent recovers an original designation. Furthermore, the main claims were indeed established by \textit{H. Tagawa} [RIMS Kôkyûroku 765, 91--103 (1991; Zbl 0782.17001)], [Tokyo J. Math. 16, 311--320 (1993; Zbl 0792.05009)] or \textit{M. Noumi}, \textit{Y. Masatoshi}, and \textit{K. Mimachi} [Jap. J. Math., New Ser. 19, 31--80 (1993; Zbl 0806.17016)], among others, or are easy consequences of them.
scientific article; zbMATH DE number 6825619
scientific article