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تسجيل مستخدم جديدPattern avoidance and quasisymmetric functions
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تأليف
Zachary Hamaker
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Given a set of permutations Pi, let S_n(Pi) denote the set of permutations in the symmetric group S_n that avoid every element of Pi in the sense of pattern avoidance. Given a subset S of {1,...,n-1}, let F_S be the fundamental quasisymmetric function indexed by S. Our object of study is the generating function Q_n(Pi) = sum F_{Des sigma} where the sum is over all sigma in S_n(Pi) and Des sigma is the descent set of sigma. We characterize those Pi contained in S_3 such that Q_n(Pi) is symmetric or Schur nonnegative for all n. In the process, we show how each of the resulting Pi can be obtained from a theorem or conjecture involving more general sets of patterns. In particular, we prove results concerning symmetries, shuffles, and Knuth classes, as well as pointing out a relationship with the arc permutations of Elizalde and Roichman. Various conjectures and questions are mentioned throughout.
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اقرأ أيضاً
Let S_n be the nth symmetric group. Given a set of permutations Pi we denote by S_n(Pi) the set of permutations in S_n which avoid Pi in the sense of pattern avoidance. Consider the generating function Q_n(Pi) = sum_pi F_{Des pi} where the sum is ove
r all pi in S_n(Pi) and F_{Des pi} is the fundamental quasisymmetric function corresponding to the descent set of pi. Hamaker, Pawlowski, and Sagan introduced Q_n(Pi) and studied its properties, in particular, finding criteria for when this quasisymmetric function is symmetric or even Schur nonnegative for all n >= 0. The purpose of this paper is to continue their investigation answering some of their questions, proving one of their conjectures, as well as considering other natural questions about Q_n(Pi). In particular we look at Pi of small cardinality, superstandard hooks, partial shuffles, Knuth classes, and a stability property.
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