
Christopher Coscia

Jonathan DeWitt
Keywords:
Permutations, Words, Permutation patterns, Transfer matrices, Asymptotics, Generating functions, Integer partitions
Abstract
We introduce some new classes of words and permutations characterized by the second difference condition $\pi(i1) + \pi(i+1)  2\pi(i) \leq k$, which we call the $k$convexity condition. We demonstrate that for any sized alphabet and convexity parameter $k$, we may find a generating function which counts $k$convex words of length $n$. We also determine a formula for the number of 0convex words on any fixedsize alphabet for sufficiently large $n$ by exhibiting a connection to integer partitions. For permutations, we give an explicit solution in the case $k = 0$ and show that the number of 1convex and 2convex permutations of length $n$ are $\Theta(C_1^n)$ and $\Theta(C_2^n)$, respectively, and use the transfer matrix method to give tight bounds on the constants $C_1$ and $C_2$. We also providing generating functions similar to the the continued fraction generating functions studied by Odlyzko and Wilf in the "coins in a fountain" problem.