Average norms of polynomials

In this paper we study the average \NL_{2\alpha}-norm over $T$-polynomials, where $\alpha$ is a positive integer. More precisely, we present an explicit formula for the average \NL_{2\alpha}-norm over all the polynomials of degree exactly $n$ with coefficients in $T$, where $T$ is a finite set of complex numbers, $\alpha$ is a positive integer, and $n\geq0$. In particular, we give a complete answer for the cases of Littlewood polynomials and polynomials of a given height. As a consequence, we derive all the previously known results for this kind of problems, as well as many new results.Comment: 13 pages, key words: Littlewood polynomials, Polynomials of height $h

Counting peaks at height k in a Dyck path

A Dyck path is a lattice path in the plane integer lattice $\mathbb{Z}\times\mathbb{Z}$ consisting of steps (1,1) and (1,-1), which never passes below the x-axis. A peak at height k on a Dyck path is a point on the path with coordinate y=k that is immediately preceded by a (1,1) step and immediately followed by a (1,-1) step. In this paper we find an explicit expression to the generating function for the number of Dyck paths starting at (0,0) and ending at (2n,0) with exactly r peaks at height k. This allows us to express this function via Chebyshev polynomials of the second kind and generating function for the Catalan numbers.Comment: 7 pages, 3 figure

Restricted 132-Dumont permutations

A permutation $\pi$ is said to be {\em Dumont permutations of the first kind} if each even integer in $\pi$ must be followed by a smaller integer, and each odd integer is either followed by a larger integer or is the last element of $\pi$ (see, for example, \cite{Z}). In \cite{D} Dumont showed that certain classes of permutations on $n$ letters are counted by the Genocchi numbers. In particular, Dumont showed that the $(n+1)$st Genocchi number is the number of Dummont permutations of the first kind on $2n$ letters. In this paper we study the number of Dumont permutations of the first kind on $n$ letters avoiding the pattern 132 and avoiding (or containing exactly once) an arbitrary pattern on $k$ letters. In several interesting cases the generating function depends only on $k$.Comment: 12 page