The largest and the smallest fixed points of permutations

We give a new interpretation of the derangement numbers d_n as the sum of the values of the largest fixed points of all non-derangements of length n-1. We also show that the analogous sum for the smallest fixed points equals the number of permutations of length n with at least two fixed points. We provide analytic and bijective proofs of both results, as well as a new recurrence for the derangement numbers.Comment: 7 page

A simple and unusual bijection for Dyck paths and its consequences

In this paper we introduce a new bijection from the set of Dyck paths to itself. This bijection has the property that it maps statistics that appeared recently in the study of pattern-avoiding permutations into classical statistics on Dyck paths, whose distribution is easy to obtain. We also present a generalization of the bijection, as well as several applications of it to enumeration problems of statistics in restricted permutations.Comment: 13 pages, 8 figures, submitted to Annals of Combinatoric

The Run Transform

We consider the transform from sequences to triangular arrays defined in terms of generating functions by f(x) -> (1-x)/(1-xy) f(x(1-x)/(1-xy)). We establish a criterion for the transform of a nonnegative sequence to be nonnegative, and we show that the transform counts certain classes of lattice paths by number of "pyramid ascents", as well as certain classes of ordered partitions by number of blocks that consist of increasing consecutive integers.Comment: 18 page