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