k-flaw Preference Sets

TL;DR

This paper introduces a combinatorial approach to enumerate k-flaw preference sets using bijections with labeled rooted forests, providing formulas, recurrence relations, and generating functions for these sets.

Contribution

It presents a novel combinatorial method and bijective results for counting k-flaw preference sets, expanding understanding of their enumeration and structural properties.

Findings

Derived formulas and recurrence relations for enumeration sequences.

Established bijections between k-flaw preference sets and labeled rooted forests.

Provided generating functions for the sequences of interest.

Abstract

In this paper, let $\mathcal{P}_{n;\leq s;k}^l$ denote a set of $k$-flaw preference sets $(a_1,...,a_n)$ with $n$ parking spaces satisfying that $1\leq a_i\leq s$ for any $i$ and $a_1=l$ and $p_{n;\leq s;k}^l=|\mathcal{P}_{n;\leq s;k}^l|$. We use a combinatorial approach to the enumeration of $k$-flaw preference sets by their leading terms. The approach relies on bijections between the $k$-flaw preference sets and labeled rooted forests. Some bijective results between certain sets of $k$-flaw preference sets of distinct leading terms are also given. We derive some formulas and recurrence relations for the sequences $p_{n;\leq s;k}^l$ and give the generating functions for these sequences.