Cycles and sorting index for matchings and restricted permutations

We prove that the Mahonian-Stirling pairs of permutation statistics (\sor, \cyc) and (\inv, \mathrm{rlmin}) are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters. Moreover, we define a sorting index for bicolored matchings and use it to show analogous equidistribution results for restricted permutations of type $B_n$ and $D_n$.Comment: 23 page

Generalizations of Permutation Statistics to Words and Labeled Forests

A classical result of MacMahon shows the equidistribution of the major index and inversion number over the symmetric groups. Since then, these statistics have been generalized in many ways, and many new permutation statistics have been defined, which are related to the major index and inversion number in may interesting ways. In this dissertation we study generalizations of some newer statistics over words and labeled forests. Foata and Zeilberger defined the graphical major index, majU , and the graphical inversion index, invU , for words over the alphabet {1, . . . , n}. In this dissertation we define a graphical sorting index, sorU , which generalizes the sorting index of a permutation. We then characterize the graphs U for which sorU is equidistributed with invU and majU on a single rearrangement class. Bj¨orner and Wachs defined a major index for labeled plane forests, and showed that it has the same distribution as the number of inversions. We define and study the distributions of a few other natural statistics on labeled forests. Specifically, we introduce the notions of bottom-to-top maxima, cyclic bottom-to-top maxima, sorting index, and cycle minima. Then we show that the pairs (inv, BT-max), (sor, Cyc), and (maj, CBT-max) are equidistributed. Our results extend the result of Bj¨orner and Wachs and generalize results for permutations. Lastly, we study the descent polynomial of labeled forests. The descent polynomial for per-mutations is known to be log-concave and unimodal. In this dissertation we discuss what properties are preserved in the descent polynomial of labeled forests

Counting derangements with signed right-to-left minima and excedances

Recently Alexandersson and Getachew proved some multivariate generalizations of a formula for enumerating signed excedances in derangements. In this paper we first relate their work to a recent continued fraction for permutations and confirm some of their observations. Our second main result is two refinements of their multivariate identities, which clearly explain the meaning of each term in their main formulas. We also explore some similar formulas for permutations of type B.Comment: Advances in Applied Mathematics 152, 10259

Colored de Bruijn Graphs and the Genome Halving Problem

Breakpoint graph analysis is a key algorithmic technique in studies of genome rearrangements. However, breakpoint graphs are defined only for genomes without duplicated genes, thus limiting their applications in rearrangement analysis. We discuss a connection between the breakpoint graphs and de Bruijn graphs that leads to a generalization of the notion of breakpoint graph for genomes with duplicated genes. We further use the generalized breakpoint graphs to study the Genome Halving Problem (first introduced and solved by Nadia El-Mabrouk and David Sankoff). The El-Mabrouk-Sankoff algorithm is rather complex, and, in this paper, we present an alternative approach that is based on generalized breakpoint graphs. The generalized breakpoint graphs make the El-Mabrouk-Sankoff result more transparent and promise to be useful in future studies of genome rearrangements