k-flaw Preference Sets

Abstract

In this paper, let ${\mathcal{P}}_{n;\le s;k}^l$ denote a set of $k$-flaw preference sets $(a_1,...,a_n)$ with $n$ parking spaces satisfying that $1\le a_i\le s$ for any $i$ and $a_1=l$ and $p_{n;\le s;k}^l=|{\mathcal{P}}_{n;\le s;k}^l|$. 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 $p_{n;\le s;k}^l$ and give the generating functions for these sequences.

33 pages

References in corpus (1)

• Counting Defective Parking Functions