Maximal 0-1 fillings of moon polyominoes with restricted chain-lengths and rc-graphs

We show that maximal 0-1-fillings of moon polynomials with restricted chain lengths can be identified with certain rc-graphs, also known as pipe dreams. In particular, this exhibits a connection between maximal 0-1-fillings of Ferrers shapes and Schubert polynomials. Moreover, it entails a bijective proof showing that the number of maximal fillings of a stack polyomino S with no north-east chains longer than k depends only on k and the multiset of column heights of S. Our main contribution is a slightly stronger theorem, which in turn leads us to conjecture that the poset of rc-graphs with covering relation given by generalised chute moves is in fact a lattice.Comment: 22 pages, v2: references added, v3: included proof for bijection for stack polyominoes, v4: include conjecture and improve presentatio

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

Multi-triangulations as complexes of star polygons

Maximal $(k+1)$-crossing-free graphs on a planar point set in convex position, that is, $k$-triangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of looking at $k$-triangulations, namely as complexes of star polygons. With this tool we give new, direct, proofs of the fundamental properties of $k$-triangulations, as well as some new results. This interpretation also opens-up new avenues of research, that we briefly explore in the last section.Comment: 40 pages, 24 figures; added references, update Section

Depth and regularity of tableau ideals

We compute the depth and regularity of ideals associated with arbitrary fillings of positive integers to a Young diagram, called the tableau ideals

Monotone Sequences in Combinatorial Structures

Symmetry of monotone sequences arise in many combinatorial structures, the classical examples being inversions and coinversions in permutations. Another example is crossings and nestings in matchings, partitions and permutations. These examples can be generalized to fillings of Ferrers diagrams and further generalized to moon polyominoes. This dissertation first introduces layer polyominoes, then extends the joint symmetry between northeast and southeast chains exhibited in moon polyominoes. For a given structure it’s not always true that symmetry of crossings and nestings holds. We introduce a type of matching, called an alternating matching, where the distribution of crossings and nestings is not symmetric. We prove a necessary and sufficient condition for an alternating matching to be non-nesting and use this to partially enumerate non-nesting alternating matchings. Finally, we prove several results on crossings and nestings in graphs. First we show that the crossing number and nesting numbers are unrelated, i.e. there are families of graphs with no crossings and with nestings numbers that diverge and vice versa. Second we give a bijection between plane trees and bi-colored motzkin paths. Lastly, we provide a generating function for a special class of Ferrers diagrams, where each row a fixed length shorter than the previous row, and the filling of the diagram has no southeast chains