论文标题
通过加泰罗尼亚矩阵的部分排列的统计数据
Statistics of Partial Permutations via Catalan matrices
论文作者
论文摘要
一个通用的Catalan矩阵$（a_ {n，k}）_ {n，k \ ge 0} $由两个种子序列$ \ mathbf {s} =（s_0，s_1，\ ldots）$和$ \ mathbf {s_0，s_1，s_1，\ ldots）$和$ \ mathbf {t} =（t_1，t_1，t_1，t_2，t_2，\ ldots）一起。通过服用$ s_ \ ell = 2 \ ell+1 $和$ t_ \ ell = \ ell^2 $，我们可以将$ a_ {n，k} $解释为部分排列的数量，即$ n \ times n $ $ 0,1 $ 0,1 $ - $ k $ ZERO的行，最多在每行或列中，最多为1 $ $ 1 $。在本文中，我们证明，关于排列的大多数基本统计数据和某些设定值的统计数据也可以根据部分排列定义，并在种子序列中进行编码。还给出了两个有趣的置换家族的结果，即连接的排列和周期上下排列。
A generalized Catalan matrix $(a_{n,k})_{n,k\ge 0}$ is generated by two seed sequences $\mathbf{s}=(s_0,s_1,\ldots)$ and $\mathbf{t}=(t_1,t_2,\ldots)$ together with a recurrence relation. By taking $s_\ell=2\ell+1$ and $t_\ell=\ell^2$ we can interpret $a_{n,k}$ as the number of partial permutations, which are $n\times n$ $0,1$-matrices of $k$ zero rows with at most one $1$ in each row or column. In this paper we prove that most of fundamental statistics and some set-valued statistics on permutations can also be defined on partial permutations and be encoded in the seed sequences. Results on two interesting permutation families, namely the connected permutations and cycle-up-down permutations, are also given.
