Enumerating Anchored Permutations with Bounded Gaps

TL;DR

This paper studies anchored permutations with bounded gaps, proving their generating functions are rational, deriving recursive formulas for specific cases, and analyzing their asymptotic growth.

Contribution

It establishes the rationality of generating functions for anchored k-bounded permutations and provides explicit recursive formulas for k=2 and 3, resolving a known conjecture.

Findings

Generating functions are always rational.

Recursive formulas for k=2 and 3 are derived.

Number of permutations grows asymptotically as O(k^n).

Abstract

Say that a permutation of $1,2,\ldots,n$ is \textit{$k$-bounded} if every pair of consecutive entries in the permutation differs by no more than $k$. Such a permutation is \textit{anchored} if the first entry is $1$ and the last entry is $n$. We show that the generating function for the enumeration of $k$-bounded anchored permutations is always rational, mirroring the known result on (non-anchored) $k$-bounded permutations due to Avgustinovich and Kitaev. We then explicitly determine the recursive formulas of minimal depth for the number of anchored $k$-bounded permutations of $n$ for $k=2$ and $k=3$, resolving a conjecture listed on the Online Encyclopedia of Integer Sequences (entry A249665). We additionally show that the number of anchored $k$-bounded permutations of $n$ is asymptotically $O\left(k^n\right)$ as a function of $n$ for a given $k$.