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We present bijections between four classes of combinatorial objects. Two of them, the class of unlabeled (2 + 2)-free posets and a certain class of involutions (or chord diagrams), already appeared in the literature, but were apparently not known to be equinumerous. We present a direct bijection between them. The third class is a family of permutations… (More)

This paper presents a bijection between ascent sequences and upper triangular matrices whose non-negative entries are such that all rows and columns contain at least one non-zero entry. We show the equivalence of several natural statistics on these structures under this bijec-tion and prove that some of these statistics are equidistributed. Several special… (More)

Motivated by juggling sequences and bubble sort, we examine permutations on the set {1, 2,. .. , n} with d descents and maximum drop size k. We give explicit formulas for enumerating such permutations for given integers k and d. We also derive the related generating functions and prove unimodality and symmetry of the coefficients. Résumé. Motivés par les "… (More)

In this paper we present a bijection between composition matrices and (2 + 2)-free posets. This bijection maps partition matrices to factorial posets, and induces a bijection from upper triangular matrices with non-negative entries having no rows or columns of zeros to unlabeled (2 + 2)-free posets. Chains in a (2 + 2)-free poset are shown to correspond to… (More)

- Mark Dukes, Astrid Reifegerste
- 2008

In this paper we study a mapping from permutations to Dyck paths. A Dyck path gives rise to a (Young) diagram and we give relationships between statistics on permutations and statistics on their corresponding diagrams. The expectation and variance of the size of the diagram are also given. We propose a filling of the diagram which determines the permutation… (More)

- Anders Claesson, Mark Dukes, Sergey Kitaev
- 2010

In connection with Vassiliev's knot invariants, Stoimenow (1998) introduced certain matchings, also called regular linearized chord diagrams. Bousquet-Mélou et al. (2008) gave a bijection from those matchings to unlabeled (2 + 2)-free posets; they also showed how to encode the posets as so called ascent sequences. In this paper we present a direct encoding… (More)

- Jean-Christophe Aval, Michele D 'adderio, Mark Dukes, Angela Hicks, Yvan Le Borgne
- 2013

We study the statistics area, bounce and dinv on the set of parallelogram polyominoes having a rectangular m times n bounding box. We show that the bi-statistics (area, bounce) and (area, dinv) give rise to the same q, t-analogue of Narayana numbers which was introduced by two of the authors in [4]. We prove the main conjectures of that paper: the q,… (More)

- Mark Dukes
- 2012

In this talk I will highlight some results from a recent paper (arXiv:1208.0024) that was motived by a correspondence between bivin-cular permutation patterns and composition matrices. We study recurrent configurations of the sandpile model on the complete bipartite graph K m,n and show how they can be classified in terms of a class of polyominoes. A… (More)