CanaDAM 2019
SFU Harbour Centre, May 29 - 31, 2019

Counting and enumeration
Chair: Joe Sawada (University of Guelph)

KAREL KLOUDA, Czech Technical University in Prague
Synchronizing delay of (epi)Sturmian morphisms  [PDF]

The synchronizing delay (SD) of a morphism is a constant strongly related to essential properties of the language generated by the morphism: morphisms with finite SD (i.e. circular/recognizable morphisms) are known not to contain arbitrarily long repetitions and to have a regular structure of bispecial factors determining the language complexity. An algorithm deciding finiteness of the SD is known. However, there is no known universal upper bound on the value of SD, and its actual value has been computed only for most simple classes of morphisms. We have found an effective upper bound for the class of Sturmian morphisms (attained by standard Sturmian morphisms) and also an upper bound for a more general class of episturmian morphisms: The SD of a primitive episturmian morphism $\varphi$ over an alphabet $\mathcal{A}$ is less than $$ \frac{1}{\#\mathcal{A} -1} \left( \sum_{a\, \in \mathcal{A}} |\varphi(a)| - 1 \right) + \max_{a\, \in \mathcal{A}}|\varphi(a)| -3\,. $$

SAMUEL SIMON, Simon Fraser University
The asymptotics of reflectable weighted walks in arbitrary dimension  [PDF]

A walk on the square lattice is a sequence of steps from a given step set, and the length of the walks is the size of the sequence. The enumeration and asymptotics of walks has been of interest, and much progress has been made within the past few decades. We look at a particular weighted family of walks confined to the positive orthant in $d$ dimensions.

We introduce the standard techniques used in manipulating functional equations to extract the desired terms from generating functions. Then we combine results from analytic combinatorics and complex analysis to find asymptotics.

FOSTER TOM, University of California, Berkeley
Classifying the near-equality of ribbon Schur functions  [PDF]

We consider the problem of determining when the difference of two ribbon Schur functions is a single Schur function. We prove that this near-equality phenomenon occurs in sixteen infinite families and we conjecture that these are the only possible cases. Towards this converse, we prove that under certain additional assumptions the only instances of near-equality are among our sixteen families. In particular, we prove that our first ten families are a complete classification of all cases where the difference of two ribbon Schur functions is a single Schur function whose corresponding partition has at most two parts at least 2. We then provide a framework for interpreting the remaining six families and we explore some ideas towards resolving our conjecture in general. We also determine some necessary conditions for the difference of two ribbon Schur functions to be Schur-positive.

AM√ČLIE TROTIGNON, Simon Fraser University & Institut Denis Poisson, Universit√© de Tours
Walks avoiding a quadrant  [PDF]

Enumeration of planar lattice walks in cones has many applications in combinatorics and probability theory. The objects are amenable to treatment by many techniques: combinatorics, complex analysis, probability theory, computer algebra, and Galois theory of difference equations. Walks restricted to the first quadrant are well studied but the case of three quadrants has been approached only recently. In this talk, we generalize the analytic method of walks in a quarter plane to walks in the three quarter plane. This method is composed of three main steps: write a functional equation that the generating function of walks satisfies, transform it to a boundary value problem and solve this problem. The result is a contour-integral expression for the generating function. The advantage of this method is to provide a uniform treatment for the study of walks.