Locally Convex Words and Permutations

TL;DR

This paper introduces new classes of words and permutations based on a second difference convexity condition, providing generating functions, explicit formulas, and asymptotic bounds for their enumeration.

Contribution

It defines the $k$-convexity condition for words and permutations, derives generating functions, and establishes asymptotic growth rates for specific cases, connecting to integer partitions.

Findings

Explicit formula for 0-convex words on fixed alphabets

Asymptotic bounds for 1-convex and 2-convex permutations

Generating functions similar to continued fractions

Abstract

We introduce some new classes of words and permutations characterized by the second difference condition $\pi(i-1) + \pi(i+1) - 2\pi(i) \leq k$, 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 0-convex words on any fixed-size 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 1-convex and 2-convex 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.