Chain enumeration of kk-divisible noncrossing partitions of classical types

We give combinatorial proofs of the formulas for the number of multichains in the $k$-divisible noncrossing partitions of classical types with certain conditions on the rank and the block size due to Krattenthaler and M{\"u}ller. We also prove Armstrong's conjecture on the zeta polynomial of the poset of $k$-divisible noncrossing partitions of type $A$ invariant under a $180^\circ$ rotation in the cyclic representation.Comment: 23 pages, 9 figures, final versio

Increasing and Decreasing Sequences of Length Two in 01-Fillings of Moon Polyominoes

We put recent results on the symmetry of the joint distribution of the numbers of crossings and nestings of two edges over matchings, set partitions and linked partitions, in the larger context of the enumeration of increasing and decreasing chains of length 2 in fillings of moon polyominoes.Comment: It is a updated version of a preprint entitled "On the symmetry of ascents and descents over 01-fillings of moon polyominoes". 19 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 $0$-$1$-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 $0$-$1$-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

Parking functions, labeled trees and DCJ sorting scenarios

In genome rearrangement theory, one of the elusive questions raised in recent years is the enumeration of rearrangement scenarios between two genomes. This problem is related to the uniform generation of rearrangement scenarios, and the derivation of tests of statistical significance of the properties of these scenarios. Here we give an exact formula for the number of double-cut-and-join (DCJ) rearrangement scenarios of co-tailed genomes. We also construct effective bijections between the set of scenarios that sort a cycle and well studied combinatorial objects such as parking functions and labeled trees.Comment: 12 pages, 3 figure