486 research outputs found
Two-sided permutation statistics via symmetric functions
Given a permutation statistic , define its inverse
statistic by
. We give a general approach,
based on the theory of symmetric functions, for finding the joint distribution
of and whenever
and are descent statistics:
permutation statistics that depend only on the descent composition. We apply
this method to a number of descent statistics, including the descent number,
the peak number, the left peak number, the number of up-down runs, and the
major index. Perhaps surprisingly, in many cases the polynomial giving the
joint distribution of and can
be expressed as a simple sum involving products of the polynomials giving the
(individual) distributions of and
. Our work leads to a rederivation of Stanley's
generating function for doubly alternating permutations, as well as several
conjectures concerning real-rootedness and -positivity.Comment: 43 page
-adic expansions related to continued fractions (Natural extension of arithmetic algorithms and S-adic system)
"Natural extension of arithmetic algorithms and S-adic system". July 20~24, 2015. edited by Shigeki Akiyama. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We consider S-adic expansions associated with continued fraction algorithms, where an S-adic expansion corresponds to an infinite composition of substitutions. Recall that a substitution is a morphism of the free monoid. We focus in particular on the substitutions associated with regular continued fractions (Sturmian substitutions), and with Arnoux-Rauzy, Brun, and Jacobi{Perron (multidimensional) continued fraction algorithms. We also discuss the spectral properties of the associated symbolic dynamical systems under a Pisot type assumption
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