有序k缺陷偏好集
Ordered k-flaw Preferences Sets
摘要：
在本文中，我们专注于有序$k$ -缺陷偏好集。令$\mathcal{OP}_{n,\geq k}$表示具有至少$k$缺陷和$\mathcal{S}_{n,k}=\{(x_1,...,x_{n-k})\mid x_1+x_2+... +x_{n-k}=n+k, x_i\in\mathbb{N}\}$的长度为$n$的有序偏好集的集合。我们从集合$\mathcal{OP}_{n,\geq k}$到$\mathcal{S}_{n,k}$得到一个双射。令$\mathcal{OP}_{n,k}$表示长度为$n$且恰好具有$k$缺陷的有序偏好集的集合。$(n,k)$ - \emph{flaw path}是一条从$(0,0)$开始到$(2n,0)$结束的格路径，只有两种步骤-上升步骤：$U=(1,1)$和下降步骤：$D=(1,-1)$位于$y = -k$线上并接触这条线。令$\mathcal{D}_{n,k}$表示$(n, k)$缺陷路径的集合。我们还建立了集$\mathcal{OP}_{n,k}$和$\mathcal{D}_{n,k}$之间的双射。令$op_{n,\geq k,\leq l}^m$ $(op_{n, k, =l}^m)$分别表示对于任何$i$ $(\max\{a_i\mid 1\leq i\leq n\}=l)$，具有至少$k$（精确）缺陷和前导项$m$满足$a_i\leq l$的偏好集$α=(a_1,...,a_n)$的数量。利用这些双射，我们得到了$op_{n,\geq k,\leq l}^m$的显式公式。进一步给出了$op_{n, k, =l}^m$的显式表达式。本文给出了由长度为$n$的有序$k$ -缺陷偏好集所构成的序列与首项为$m$的序列的递推关系。利用这些递推关系，我们得到了相应的$k$ -缺陷偏好集的生成函数.
英文摘要：
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 $α=(a_1,...,a_n)$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
