We enumerate involutions avoiding any single pattern of length three according to the statistic ‘number of visible pairs’, namely, the number of non-adjacent columns in the bargraph representation of a given permutation that is mutually visible to each other. The proofs are combinatorial and reside on well-known bijections between pattern avoiding involutions and lattice paths.

中文翻译：

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受限对合中的可见性

我们根据统计“可见对数”枚举对合，避免任何长度为 3 的单一模式，即给定排列的条形图表示中相互可见的非相邻列的数量。证明是组合的，并且存在于模式避免对合和晶格路径之间的众所周知的双射上。