Continued fractions for permutation statistics

We explore a bijection between permutations and colored Motzkin paths that has been used in different forms by Foata and Zeilberger, Biane, and Corteel. By giving a visual representation of this bijection in terms of so-called cycle diagrams, we find simple translations of some statistics on permutations (and subsets of permutations) into statistics on colored Motzkin paths, which are amenable to the use of continued fractions. We obtain new enumeration formulas for subsets of permutations with respect to fixed points, excedances, double excedances, cycles, and inversions. In particular, we prove that cyclic permutations whose excedances are increasing are counted by the Bell numbers.Comment: final version formatted for DMTC

Enumeration of Standard Young Tableaux

A survey paper, to appear as a chapter in a forthcoming Handbook on Enumeration.Comment: 65 pages, small correction

Enumerating two permutation classes by number of cycles

We enumerate permutations in the two permutation classes $\text{Av}_n(312, 4321)$ and $\text{Av}_n(321, 4123)$ by the number of cycles each permutation admits. We also refine this enumeration with respect to several statistics

Analysis of casino shelf shuffling machines

Many casinos routinely use mechanical card shuffling machines. We were asked to evaluate a new product, a shelf shuffler. This leads to new probability, new combinatorics and to some practical advice which was adopted by the manufacturer. The interplay between theory, computing, and real-world application is developed.Comment: Published in at http://dx.doi.org/10.1214/12-AAP884 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Restricted Dumont permutations, Dyck paths, and noncrossing partitions

We complete the enumeration of Dumont permutations of the second kind avoiding a pattern of length 4 which is itself a Dumont permutation of the second kind. We also consider some combinatorial statistics on Dumont permutations avoiding certain patterns of length 3 and 4 and give a natural bijection between 3142-avoiding Dumont permutations of the second kind and noncrossing partitions that uses cycle decomposition, as well as bijections between 132-, 231- and 321-avoiding Dumont permutations and Dyck paths. Finally, we enumerate Dumont permutations of the first kind simultaneously avoiding certain pairs of 4-letter patterns and another pattern of arbitrary length.Comment: 20 pages, 5 figure