Pattern avoidance in matchings and partitions
Abstract
Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards. We enumerate 312-avoiding matchings and partitions, obtaining algebraic generating functions, in contrast with the known D-finite generating functions for the 321-avoiding (i.e., 3-noncrossing) case. Our approach also provides a more direct proof of a formula of Bóna for the number of 1342-avoiding permutations. Additionally, we give a bijection proving the shape-Wilf-equivalence of the patterns 321 and 213 which greatly simplifies existing proofs by Backelin--West--Xin and Jelínek, and provides an extension of work of Gouyou-Beauchamps for matchings with fixed points. Finally, we classify pairs of patterns of length 3 according to shape-Wilf-equivalence, and enumerate matchings and partitions avoiding a pair in most of the resulting equivalence classes.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2012
- DOI:
- 10.48550/arXiv.1211.3442
- arXiv:
- arXiv:1211.3442
- Bibcode:
- 2012arXiv1211.3442B
- Keywords:
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- Mathematics - Combinatorics;
- 05A15 (Primary);
- 05A05;
- 05A18;
- 05A19 (Secondary)
- E-Print:
- 34 pages, 12 Figures, 5 Tables