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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)
A poset is said to be (2 + 2)-free if it does not contain an induced subposet that is isomorphic to 2 + 2, the union of two disjoint 2-element chains. In a recent paper, Bousquet-Mélou et al. found, using so called ascent sequences, the generating function for the number of (2 + 2)-free posets: P (t) = n≥0 n i=1 1 − (1 − t) i. We extend this result by(More)
In this paper we begin the first systematic study of distributions of quadrant marked mesh patterns. Mesh patterns were introduced recently by Brändén and Claesson in connection with permutation statistics. Quadrant marked mesh patterns are based on how many elements lie in various quadrants of the graph of a permutation relative to the coordinate system(More)
We introduced the notation of a set of prohibitions and give definitions of a complete set and a crucial word with respect to a given set of prohibitions. We consider 3 particular sets which appear in different areas of mathematics and for each of them examine the length of a crucial word. One of these sets is proved to be incomplete. The problem of(More)