Equidistributed Statistics on Matchings and Permutations

We show that the bistatistic of right nestings and right crossings in matchings without left nestings is equidistributed with the number of occurrences of two certain patterns in permutations, and furthermore that this equidistribution holds when refined to positions of these statistics in matchings and permutations. For this distribution we obtain a non-commutative generating function which specializes to Zagier's generating function for the Fishburn numbers after abelianization.As a special case we obtain proofs of two conjectures of Claesson and Linusson.Finally, we conjecture that our results can be generalized to involving left crossings of matchings too.

Equivalence Classes of Permutations Modulo Replacements Between 123 and Two-Integer Patterns

We explore a new type of replacement of patterns in permutations, suggested by James Propp, that does not preserve the length of permutations. In particular, we focus on replacements between 123 and a pattern of two integer elements. We apply these replacements in the classical sense; that is, the elements being replaced need not be adjacent in position or value. Given each replacement, the set of all permutations is partitioned into equivalence classes consisting of permutations reachable from one another through a series of bi-directional replacements. We break the eighteen replacements of interest into four categories by the structure of their classes and fully characterize all of their classes.