k-flaw Preference Sets
  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. 
Po-Yi Huang, Jun Ma, Jean Yeh