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A cyclic polytope of dimension $d$ with $n$ vertices is a convex polytope combinatorially equivalent to the convex hull of $n$ distinct points on a moment curve in ${\Bbb R}^d$. In this paper, we prove the cyclic sieving phenomenon, introduced by Reiner-Stanton-White, for faces of an even-dimensional cyclic polytope, under a group action that cyclically translates the vertices. For odd-dimensional cyclic polytopes, we enumerate the faces that are invariant under an automorphism that reverses the order of the vertices and an automorphism that interchanges the two end vertices, according to the order on the curve. In particular, for $n=d+2$, we give instances of the phenomenon under the groups that cyclically translate the odd-positioned and even-positioned vertices, respectively.

Let $\Gamma$ denote a distance-regular graph with diameter $D \geq 3$ and intersection numbers $a_1=0, a_2 \neq 0$, and $c_2=1$. We show a connection between the $d$-bounded property and the nonexistence of parallelograms of any length up to $d+1$. Assume further that $\Gamma$ is with classical parameters $(D, b, \alpha, \beta)$, Pan and Weng (2009) showed that $(b, \alpha, \beta)= (-2, -2, ((-2)^{D+1}-1)/3).$ Under the assumption $D \geq 4$, we exclude this class of graphs by an application of the above connection.

In this paper we present two interesting properties of 321-avoiding Baxter permutations. The first one is a variant of refined major-balance identity for the 321-avoiding Baxter permutations, respecting the number of fixed points and descents. The second one is a bijection between the 321-avoiding Baxter permutations with the entry 1 preceding the entry 2 and the positive braid words on four strands.

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