Ordered k-flaw Preferences Sets
  In this paper, we focus on ordered $k$-flaw preference sets. Let $\mathcal{OP}_{n,\geq k}$ denote the set of ordered preference sets of length $n$ with at least $k$ flaws and $\mathcal{S}_{n,k}=\{(x_1,...,x_{n-k})\mid x_1+x_2+... +x_{n-k}=n+k, x_i\in\mathbb{N}\}$. We obtain a bijection from the sets $\mathcal{OP}_{n,\geq k}$ to $\mathcal{S}_{n,k}$. Let $\mathcal{OP}_{n,k}$ denote the set of ordered preference sets of length $n$ with exactly $k$ flaws. An $(n,k)$-\emph{flaw path} is a lattice path starting at $(0,0)$ and ending at $(2n,0)$ with only two kinds of steps--rise step: $U=(1,1)$ and fall step: $D=(1,-1)$ lying on the line $y = -k$ and touching this line. Let $\mathcal{D}_{n,k}$ denote the set of $(n, k)$-flaw paths. Also we establish a bijection between the sets $\mathcal{OP}_{n,k}$ and $\mathcal{D}_{n,k}$. Let $op_{n,\geq k,\leq l}^m$ $(op_{n, k, =l}^m)$ denote the number of preference sets $\alpha=(a_1,...,a_n)$ with at least $k$ (exact) flaws and leading term $m$ satisfying $a_i\leq l$ for any $i$ $(\max\{a_i\mid 1\leq i\leq n\}=l)$, respectively. With the benefit of these bijections, we obtain the explicit formulas for $op_{n,\geq k,\leq l}^m$. Furthermore, we give the explicit formulas for $op_{n, k, =l}^m$. We derive some recurrence relations of the sequence formed by ordered $k$-flaw preference sets of length $n$ with leading term $m$. Using these recurrence relations, we obtain the generating functions of some corresponding $k$-flaw preference sets. 
Po-Yi Huang, Jun Ma, Yeong-Nan Yeh