arXiv 1306.54282013-06-23math.CO

关于具有有界滴大小的置换

On Permutations with Bounded Drop Size

Joanna N. Chen, William Y. C. Chen

备注：: 19 pages

摘要：

将$[n]=\{1,2,\ldots, n\}$的置换$π$的最大丢弃大小定义为$i-π(i)$的最大值。Chung、Claesson、Dukes和Graham获得了多项式$P_k(x)$，该多项式可用于确定$d$下降且最大液滴尺寸不大于$k$的$[n]$的排列数。此外，Chung和Graham给出了$Q_k(x)=x^k P_k(x)$和$R_{n,k}(x)=Q_k(x)(1+x+\cdots+x^k)^{n-k}$的系数的组合解释，并提出了寻找$R_{n,k}(x)$对称性的双射证明的问题。本文在$A_{n,k}$上建立了一个双射$φ$，其中$A_{n,k}$是$[n]$的置换集，最大液滴尺寸不大于$k$ .映射$φ$仍然是$A_{n,k}$的某些子集之间的双射。%与对称性有关。这为Chung和Graham的问题提供了答案。本文的第二个结果证明了Hyatt关于多项式的单峰性与$[n]$的符号置换数有关的猜想，其中$[n]$具有$d$型$B$下降且$B$型最大滴大小不大于$k$ .

英文摘要：

The maximum drop size of a permutation $π$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-π(i)$. Chung, Claesson, Dukes and Graham obtained polynomials $P_k(x)$ that can be used to determine the number of permutations of $[n]$ with $d$ descents and maximum drop size not larger than $k$. Furthermore, Chung and Graham gave combinatorial interpretations of the coefficients of $Q_k(x)=x^k P_k(x)$ and $R_{n,k}(x)=Q_k(x)(1+x+\cdots+x^k)^{n-k}$, and raised the question of finding a bijective proof of the symmetry property of $R_{n,k}(x)$. In this paper, we establish a bijection $φ$ on $A_{n,k}$, where $A_{n,k}$ is the set of permutations of $[n]$ and maximum drop size not larger than $k$. The map $φ$ remains to be a bijection between certain subsets of $A_{n,k}$. %related to the symmetry property. This provides an answer to the question of Chung and Graham. The second result of this paper is a proof of a conjecture of Hyatt concerning the unimodality of polynomials in connection with the number of signed permutations of $[n]$ with $d$ type $B$ descents and the type $B$ maximum drop size not greater than $k$.

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。

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英文摘要：

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