7 research outputs found
Finitely labeled generating trees and restricted permutations
Generating trees are a useful technique in the enumeration of various
combinatorial objects, particularly restricted permutations. Quite often the
generating tree for the set of permutations avoiding a set of patterns requires
infinitely many labels. Sometimes, however, this generating tree needs only
finitely many labels. We characterize the finite sets of patterns for which
this phenomenon occurs. We also present an algorithm - in fact, a special case
of an algorithm of Zeilberger - that is guaranteed to find such a generating
tree if it exists.Comment: Accepted by J. Symb. Comp.; 12 page
Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes
We put recent results by Chen, Deng, Du, Stanley and Yan on crossings and
nestings of matchings and set partitions in the larger context of the
enumeration of fillings of Ferrers shape on which one imposes restrictions on
their increasing and decreasing chains. While Chen et al. work with
Robinson-Schensted-like insertion/deletion algorithms, we use the growth
diagram construction of Fomin to obtain our results. We extend the results by
Chen et al., which, in the language of fillings, are results about
--fillings, to arbitrary fillings. Finally, we point out that, very
likely, these results are part of a bigger picture which also includes recent
results of Jonsson on --fillings of stack polyominoes, and of results of
Backelin, West and Xin and of Bousquet-M\'elou and Steingr\'\i msson on the
enumeration of permutations and involutions with restricted patterns. In
particular, we show that our growth diagram bijections do in fact provide
alternative proofs of the results by Backelin, West and Xin and by
Bousquet-M\'elou and Steingr\'\i msson.Comment: AmS-LaTeX; 27 pages; many corrections and improvements of
short-comings; thanks to comments by Mireille Bousquet-Melou and Jakob
Jonsson, the final section is now much more profound and has additional
result
Matchings Avoiding Partial Patterns
We show that matchings avoiding a certain partial pattern are counted by the 3-Catalan numbers. We give a characterization of 12312-avoiding matchings in terms of restrictions on the corresponding oscillating tableaux. We also find a bijection between matchings avoiding both patterns 12312 and 121323 and Schröder paths without peaks at level one, which are counted by the super-Catalan numbers or the little Schröder numbers. A refinement of the super-Catalan numbers is derived by fixing the number of crossings in the matchings. In the sense of Wilf-equivalence, we use the method of generating trees to show that the patterns 12132, 12123, 12321, 12231, 12213 are all equivalent to the pattern 12312.
Matchings avoiding partial patterns and lattice paths
In this paper, we consider matchings avoiding partial patterns 1123 and 1132. We give a bijection between 1123-avoiding matchings with n edges and nonnegative lattice paths from (0, 2) to (2n, 0). As a consequence, the refined enumeration of 1123-avoiding matchings can be reduced to the enumeration of certain lattice paths. Another result of this paper is a bijection between 1132-avoiding matchings with n edges and lattice paths from (0, 0) to (2n, 0) starting with an up step, which may go under the x-axis.