A combinatorial proof of q-log-concavity of q-Eulerian numbers

Abstract

Carlitz established a q-analog of the Eulerian numbers $A_{n,k}(q)$ and defined the relationship $A_{n,k}(q)=q^{\frac{(n-k)(n-k+1)}{2}}A_{n,k}^{*}(q)$. In this paper, by using the combinatorial interpretation of $A_{n,k}^{*}(q)$ and constructing injective maps, we prove that $A_{n,k}^{*}(q)$ and $A_{n,k}(q)$ are q-log-concave, that is, all the coefficients of the polynomials $( A_{n,k}^{*}(q)) ^{2}- A_{n,k-1}^{*}(q) A_{n,k+1}^{*}(q)$ and $(A_{n,k}(q)) ^{2}- A_{n,k-1}(q) A_{n,k+1}(q)$ are nonnegative for $1< k <n$.

Appendix

For the lattice path $\sigma ,\ \beta ,\ w=\phi (\sigma ),$ and $v=\phi (\beta )$ sketched in Fig. 3, we obtain

$$\begin{aligned} P(q)= & {} (1+q)^{2}(1+q+q^{2})^{2}(1+q+q^{2}+q^{3})^{2}\\{} & {} \times (1+q+q^{2}+q^{3}+q^{4}),\\ Q(q)= & {} (1+q)^{2}(1+q+q^{2})^{2}(1+q+q^{2}+q^{3})^{2}\\{} & {} \times (1+q+q^{2}+q^{3}+q^{4}),\\ M(q)= & {} (1+q+q^{2})^{2}(1+q+q^{2}+q^{3})^{2}\\{} & {} \times (1+q+q^{2}+q^{3}+q^{4}+q^{5}),\\ N(q)= & {} (1+q)(1+q+q^{2})(1+q+q^{2}+q^{3})\\{} & {} \times (1+q+q^{2}+q^{3}+q^{4})\\{} & {} \times (1+q+q^{2}+q^{3}+q^{4}+q^{5})\\ P(q)Q(q)-M(q)N(q)= & {} (1+q)^{4}(1+q+q^{2})^{4}(1+q+q^{2}+q^{3})^{4}\\{} & {} \times (1+q+q^{2}+q^{3}+q^{4})^{2}-(1+q)(1+q+q^{2})^{3}\\{} & {} \times (1+q+q^{2}+q^{3})^{3}(1+q+q^{2}+q^{3}+q^{4})\\{} & {} \times (1+q+q^{2}+q^{3}+q^{4}+q^{5})^{2}\\= & {} q^{32}+14 q^{31}+100 q^{30}+490 q^{29}+1852 q^{28}\\{} & {} +5743 q^{27}+15169 q^{26}+34981 q^{25}+71658 q^{24}\\{} & {} +132037 q^{23}+220887 q^{22}+337858 q^{21}+474989 q^{20}\\{} & {} +616205 q^{19}+739747 q^{18}+823317 q^{17}\\{} & {} +850381 q^{16}+815280 q^{15}+725070 q^{14}+597307 q^{13}\\{} & {} +454692 q^{12}+318731 q^{11}+204753 q^{10}\\{} & {} +119762 q^{9}+63228 q^{8}+29777 q^{7}+12307 q^{6}\\{} & {} +4361 q^{5}+1279 q^{4}+293 q^{3}+47 q^{2}+4 q,\\ P(q)Q(q)-qM(q)N(q)= & {} (1+q)^{4}(1+q+q^{2})^{4}\\{} & {} \times (1+q+q^{2}+q^{3})^{4}(1+q+q^{2}+q^{3}+q^{4})^{2}\\ {}{} & {} -q(1+q)(1+q+q^{2})^{3}(1+q+q^{2}+q^{3})^{3}\\{} & {} \times (1+q+q^{2}+q^{3}+q^{4})(1+q+q^{2}+q^{3}+q^{4}+q^{5})^{2}\\= & {} q^{32}+13 q^{31}+91 q^{30}+446 q^{29}\\{} & {} +1699 q^{28}+5323q^{27}{+}14207 q^{26}{+}33081 q^{25}+68354 q^{24}\\{} & {} +126911 q^{23}+213738 q^{22}+328873 q^{21}\\{} & {} +464847 q^{20}+606050q^{19}+731004 q^{18}+817383 q^{17}\\{} & {} +848278 q^{16}+817383 q^{15}+731004 q^{14}\\{} & {} +606050 q^{13}+464847q^{12}+328873 q^{11}+213738 q^{10}\\{} & {} +126911 q^{9}+68354 q^{8}+33081 q^{7}\\{} & {} +14207 q^{6}+5323 q^{5}+1699q^{4}+446 q^{3}+91 q^{2}+13 q +1. \end{aligned}$$

It is clear that $P_{w}(q)P_{v}(q)-P_{\sigma }(q)P_{\beta }(q)\ge _{q} 0$ and $P_{w}(q)P_{v}(q)-qP_{\sigma }(q)P_{\beta }(q)\ge _{q} 0$.