A centrosymmetric permutation is one which is invariant under the reverse-complement operation, or equivalently one whose associated standard Young tableaux under the Robinson-Schensted algorithm are both invariant under the Schutzenberger involution. In this paper, we characterize the set of permutations avoiding 1243 and 2143 whose images under the reverse-complement mapping also avoid these patterns. We also characterize in a simple manner the corresponding Schroder paths under a bijection of Egge and Mansour. We then use these results to enumerate centrosymmetric permutations avoiding the patterns 1243 and 2143. In a similar manner, centrosymmetric involutions avoiding these same patterns are shown to be enumerated by the Pell numbers.
In this note, we study the mean length of the longest increasing subsequence of a uniformly sampled involution that avoids the pattern $3412$ and another pattern.
We enumerate and characterize some classes of alternating and reverse alternating involutions avoiding a single pattern of length three or four. If on one hand the case of patterns of length three is trivial, on the other hand, the length four case is more challenging and involves sequences of combinatorial interest, such as Motzkin and Fibonacci numbers.
Enumerating Pattern-Avoiding Involutions using Combinatorial Exploration
School of Computer Science and Mathematics, Keele University（基尔大学计算机科学与数学学院）；School of Mathematics and Statistics, The University of Melbourne（墨尔本大学数学与统计学院）；Department of Mathematical and Statistical Sciences, Marquette University（马凯特大学数学与统计科学系）
Bijections from Dyck paths to 321-avoiding permutations revisited
文献中至少有三个从Dyck路径到321-避免置换的双射，分别是Billey-Jockusch-Stanley，Krattenthaler和Mansour-Deng-Du的双射，它们有什么不同？分别记为B，K，M，证明了M = B \circ L = K \circ L'，其中L是Dyck路上的经典Kreweras-Lalanne对合，L'也是对合，是L的一种导函数.因此，K^{-1} \circ B，B和K之间的差的度量，是对合L' \circ L的乘积，结果是一个非常奇怪的双射：作为Dyck n-路上的置换，它是“反向路”对合的n次方根。这一事实的证明归结为一个几何参数涉及对不相交的格路径。
We examine the enumeration of certain Motzkin objects according to the numbers of crossings and nestings. With respect to continued fractions, we compute and express the distributions of the statistics of the numbers of crossings and nestings over three sets, namely the set of $4321$-avoiding involutions, the set of $3412$-avoiding involutions, and the set of $(321,3\bar{1}42)$-avoiding permutations. To get our results, we exploit the bijection of Biane restricted to the sets of $4321$- and $3412$-avoiding involutions which was characterized by Barnabei et al.~ and the bijection between $(321,…
We give a natural definition of rowmotion for $321$-avoiding permutations, by translating, through bijections involving Dyck paths and the Lalanne--Kreweras involution, the analogous notion for antichains of the positive root poset of type $A$. We prove that some permutation statistics, such as the number of fixed points, are homomesic under rowmotion, meaning that they have a constant average over its orbits. Our setting also provides a more natural description of the celebrated Armstrong--Stump--Thomas equivariant bijection between antichains and non-crossing matchings in types $A$ and $B$, …
Let $\mathcal{S}_n(π)$ (resp. $\mathcal{I}_n(π)$ and $\mathcal{AI}_n(π)$) denote the set of permutations (resp. involutions and alternating involutions) of length $n$ which avoid the permutation pattern $π$. For $k,m\geq 1$, Backelin-West-Xin proved that $|\mathcal{S}_n(12\cdots kτ)|= |\mathcal{S}_n(k\cdots 21τ)|$ by establishing a bijection between these two sets, where $τ= τ_1τ_2\cdots τ_m$ is an arbitrary permutation of $k+1,k+2,\ldots,k+m$. The result has been extended to involutions by Bousquet-Mélou and Steingrímsson and to alternating permutations by the first author. In this paper, we …
在本文中，我们定义和研究我们所谓的双重加泰罗尼亚么半群。这个幺半群是从0-Hecke幺半群到二元关系幺半群的自然映射的图像。我们表明，双加泰罗尼亚幺半群提供了一个代数化的（组合）集的4321-避免排列和相关的组合学的各种分支的加泰罗尼亚数的组合学和组合的排列。特别地，我们给出了Dyck路径上Kreweras对合的一阶导数、4321-避免对合和Barnabei { \em et al.}关于Dyck路径的容许对我们计算一个演示文稿，并确定一个有效的双重加泰罗尼亚幺半群表示的最小维数。我们还确定了0-Hecke幺半群的有效表示的最小维数。
