Abstract:Which combinatorial sequences correspond to moments of probability measures on the real line? We present a generating function, in the form of a continued fraction, for a fourteen-parameter family of such sequences and interpret these in terms of combinatorial statistics on the symmetric groups. Special cases include several classical and noncommutative probability laws, along with a substantial subset of the orthogonalizing measures in the $q$-Askey scheme, now given a new combinatorial interpretation in terms of elementary permutation statistics. This framework further captures a variety of interesting combinatorial sequences including, notably, the moment sequences associated to distributions of the numbers of occurrences of (classical and vincular) permutation patterns of length three. This connection between pattern avoidance and broader ideas in classical and noncommutative probability is among several intriguing new corollaries, which generalize and unify results previously appearing in the literature, while opening up new lines of inquiry. The fourteen combinatorial statistics further generalize to signed and colored permutations, and, as an infinite family of statistics, to the $k$-arrangements: permutations with $k$-colored fixed points, introduced here along with several related results and conjectures.

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中文翻译：

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排列、时刻、度量

摘要：哪些组合序列对应于实线上概率测度的矩？我们以连分数的形式为此类序列的十四参数族提供了一个生成函数，并根据对称群的组合统计来解释这些。特殊情况包括几个经典和非交换概率定律，以及 $q$-Askey 方案中正交化度量的一个重要子集，现在根据基本排列统计给出了新的组合解释。该框架进一步捕获了各种有趣的组合序列，特别是包括与长度为 3 的（经典和 vincular）置换模式的出现次数分布相关的矩序列。经典和非交换概率中的模式避免和更广泛的想法之间的这种联系是几个有趣的新推论之一，它们概括和统一了以前出现在文献中的结果，同时开辟了新的研究路线。十四个组合统计进一步推广到有符号和有色排列，并且作为一个无限的统计系列，到$k$-arrangements：带有 $k$-colored 固定点的排列，这里介绍了几个相关的结果和猜想。

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