CanaDAM 2019
SFU Harbour Centre, 29 - 31 mai 2019

Enumerative combinatorics I
Org: Sergi Elizalde (Dartmouth College, USA)

MICHAEL ALBERT, University of Otago
Wilf-equivalence and Wilf-collapse  [PDF]

When enumerative coincidences occur between collections of structures avoiding some particular substructure then they have been called Wilf-equivalences. For instance, the collection of permutations avoiding the pattern 312 is enumerated by the ubiquitous Catalan numbers. What about permutations that avoid 312 and one additional pattern of size $n$? There are $o(2.5^n)$ distinct Wilf-equivalence classes -- a Wilf-collapse. A more thorough investigation of this phenomenon leads to the conclusion that the combination of at least one non-trivial symmetry and a greedy algorithm for detecting the occurrence of a pattern leads to Wilf-collapse.

MIKLOS BONA, University of Florida
Pattern avoidance in permutations and their squares  [PDF]

We study permutations $p$ such that both $p$ and $p^2$ avoid a given pattern $q$. We will show two exact enumeration formulas, two inequalities, and present a fascinating conjecture. This is joint work with Rebecca Smith.

JAY PANTONE, Marquette University
How many chord diagrams have no short chords?  [PDF]

A chord diagram with $n$ chords is a set of $2n$ points in a line connected in n pairs. Chord diagrams, sometimes called matchings, play an important role in mathematical biology, knot theory, and combinatorics, and as a result they have been intensely studied by mathematicians, computer scientists, and biologists alike. We use a combination of symbolic, analytic, and experimental methods to examine the enumeration of chord diagrams without short chords. This is joint work with Peter Doyle and Everett Sullivan.

JESSICA STRIKER, North Dakota State University
Sign matrix polytopes from Young tableaux  [PDF]

Motivated by the study of polytopes formed as the convex hull of permutation matrices and alternating sign matrices, we define several new families of polytopes as convex hulls of \emph{sign matrices}, which are certain $\{0,1,-1\}$-matrices in bijection with semistandard Young tableaux. We discuss various properties of these polytopes, including their inequality descriptions, face lattices, and facet enumerations, as well as connections to alternating sign matrix polytopes and transportation polytopes. This is joint work with Sara Solhjem.

AE JA YEE, Penn State
A lecture hall theorem for m-falling partitions  [PDF]

One of the well-known partition theorems is Euler's theorem on partitions into distinct parts and partitions into odd parts. Recently, Keith and Xiong found a generalization of Euler's theorem for any moduli $m$. Motivated by their work, Fu, Tang and I considered a lecture hall partition analog for any moduli $m$, and we were able to prove a theorem for $m$-falling partitions. Here, an integer partition is called an $m$-falling partition if the least nonnegative residues mod $m$ of parts form a nonincreasing sequence. In this talk, I will discuss Keith-Xiong's generalization and our result.