Mark Dukes

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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)
Anastrozole is a comparatively simple, achiral benzyltriazole derivative, 2,2'-[5-(1H-1,2,4-triazol-1-ylmethyl)-1,3-phenylene]bis(2-++ +methylpropiononitrile), that inhibits human placental aromatase with an IC50 of 15 nM and elicits maximal activity in vivo in rats (inhibition of ovulation and androstenedione-induced uterine hypertrophy) and monkeys(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)
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 chord diagrams (or involutions), already appear in the literature. The third one is a class of permutations, defined in terms of a new type of pattern. An attractive property of these patterns is that, like(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)
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)
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)