2008/06/04 by Mireille Bousquet-Mélou, Mireille Bousquet‐Mélou, Anders Claesson +6 · 1 voice · 5 citations
Engineering · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems #math.CO
paper · pdf · doi:10.48550/arxiv.0806.0666
openalex publication_date 2008/06/04 · arxiv published 2008/06/04 · arxiv updated 2009/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 defined in terms of a new type of pattern. An attractive property of these patterns is that, like classical patterns, they are closed under the action of D8, the symmetry group of the square. The fourth class is formed by certain integer sequences, called ascent sequences, which have a simple recursive structure and are shown to encode (2+2)-free posets and permutations. Our bijections preserve numerous statistics. We determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for the class of chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern 3 152 4 and use this to enumerate those permutations, thereby settling a conjecture of Pudwell.