694 research outputs found
Permutations with restricted patterns and Dyck paths
We exhibit a bijection between 132-avoiding permutations and Dyck paths.
Using this bijection, it is shown that all the recently discovered results on
generating functions for 132-avoiding permutations with a given number of
occurrences of the pattern follow directly from old results on the
enumeration of Motzkin paths, among which is a continued fraction result due to
Flajolet. As a bonus, we use these observations to derive further results and a
precise asymptotic estimate for the number of 132-avoiding permutations of
with exactly occurrences of the pattern . Second,
we exhibit a bijection between 123-avoiding permutations and Dyck paths. When
combined with a result of Roblet and Viennot, this bijection allows us to
express the generating function for 123-avoiding permutations with a given
number of occurrences of the pattern in form of a continued
fraction and to derive further results for these permutations.Comment: 17 pages, AmS-Te
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
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