  The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is
defined to be the maximum value of $i-\pi(i)$. Chung, Claesson, Dukes and
Graham obtained polynomials $P_k(x)$ that can be used to determine the number
of permutations of $[n]$ with $d$ descents and maximum drop size not larger
than $k$. Furthermore, Chung and Graham gave combinatorial interpretations of
the coefficients of $Q_k(x)=x^k P_k(x)$ and
$R_{n,k}(x)=Q_k(x)(1+x+\cdots+x^k)^{n-k}$, and raised the question of finding a
bijective proof of the symmetry property of $R_{n,k}(x)$. In this paper, we
establish a bijection $\varphi$ on $A_{n,k}$, where $A_{n,k}$ is the set of
permutations of $[n]$ and maximum drop size not larger than $k$. The map
$\varphi$ remains to be a bijection between certain subsets of $A_{n,k}$.
%related to the symmetry property. This provides an answer to the question of
Chung and Graham. The second result of this paper is a proof of a conjecture of
Hyatt concerning the unimodality of polynomials in connection with the number
of signed permutations of $[n]$ with $d$ type $B$ descents and the type $B$
maximum drop size not greater than $k$.
