Some properties of the matrix related to q-coloured coordination number

Abstract

The $q$-coloured coordination number $S_{n, k}(q)$ count the number of lattice paths from (0,0) to $(n, k)$ using steps $(0,1),(1,0)$ and $(1,1)$ without east-steps on the $x$-axis, among which the $(1,1)$ steps are coloured with $q$ colours.We investigate some properties of the polynomial matrix $S(q)=\left[s_{n, k}(q)\right]_{n, k \geq 0}=\left[S_{n-k, k}(q)\right]_{n, k \geq 0}$,including the unimodality problems of sequences located over rays in $S(q)$ and the $q$-total positivity of $S(q)$.We show that the zeros of all row sums $R_n(q)=\sum_{k=0}^n s_{n, k}(q)=\sum_i r_{n, i} q^i$ are in $(-\infty,-1)$ and are dense in the corresponding semi-closed interval.We also prove that the coefficients $ r_{n, i} $ are asymptotically normal (by central and local limit theorems).