Surprising Symmetries in Objects Counted by Catalan Numbers

[摘要] We prove that the total number $S_{n,132}(q)$ of copies of  the pattern $q$ in all 132-avoiding permutations of length $n$ is the same for $q=231$, $q=312$, or $q=213$. We provide a combinatorial proof for this unexpected threefold symmetry. We then signi

[学科分类] 离散数学和组合数学

[关键词] permutations;patterns;plane trees;bijection