# One World Combinatorics on Words Seminar

The seminar takes place biweekly on Mondays at 15:00 CET. In summer, it means 6:00 in California, 9:00 in New-York or Waterloo, 14:00 in London, 15:00 in Paris, 16:00 in Moscow and 22:00 in Tokyo. In winter, some of these numbers change. Please check in advance the time in your time zone.

#### URL

The address of the Zoom meeting is https://zoom.us/j/92245493528 . The password is distributed in announcements. If you want to receive them, or receive them and want to unsubscribe, please write to Anna Frid.

All recorded talks are also available here.

#### Organizers:

Anna E. Frid, Aix-Marseille Université, Narad Rampersad, University of Winnipeg, Jeffrey O. Shallit, University of Waterloo, Manon Stipulanti, Université de Liège.

If you are interested in giving a talk, you are welcome to contact Narad Rampersad and Manon Stipulanti.

### Upcoming talks

** Feb 6 2023: Matthew Konefal** *Examining the Class of Formal Languages which are Expressible via Word Equations*

A word equation can be said to express a formal language via each variable occurring in it. The class $WE$ of formal languages which can be expressed in this way is not well understood. I will discuss a number of necessary and sufficient conditions for a formal language $L$ to belong to $WE$. I will give particular focus to the case in which $L$ is regular, and to the case in which $L$ is a submonoid.

** Feb 27 2023: Aline Parreau** and **Eric Duchêne**

** Mar 6 2023: Léo Poirier and Wolfgang Steiner**

** Mar 27 2023: Štěpán Starosta**

** April 3 2023: Jeffrey O. Shallit** *New Results in Additive Number Theory via Combinatorics on Words*

Additive number theory is the study of the additive properties of integers. Surprisingly, we can use techniques from combinatorics on words to prove results in this area. In this talk I will discuss the number of representations of an integer N as a sum of elements from some famous sets, such as the evil numbers, the odious numbers, the Rudin-Shapiro numbers, Wythoff sequences, etc.

### Past talks 2023

** Jan 23 2023: Lubomíra Dvořáková** *Essential difference between the repetitive thereshold and asymptotic repetitive threshold of balanced sequences*

At first, we will summarize both the history and the state of the art of the critical exponent and the asymptotic critical exponent of balanced sequences. Second, we will colour the Fibonacci sequence by suitable constant gap sequences to provide an upper bound on the asymptotic repetitive threshold of $d$-ary balanced sequences. The bound is attained for $d$ equal to $2$, $4$ and $8$ and we conjecture that it happens for infinitely many even $d$'s. Finally, we will reveal an essential difference in behavior of the repetitive threshold and the asymptotic repetitive threshold of balanced sequences: The repetitive threshold of $d$-ary balanced sequences is known to be at least $1+1/(d-2)$ for each $d$ larger than two. In contrast, our bound implies that the asymptotic repetitive threshold of $d$-ary balanced sequences is at most $1+\phi^3/2^{d-3}$ for each $d$ larger than one, where $\phi$ is the golden mean.

Joint work with Edita Pelantová.

** Jan 9: Pamela Fleischmann** *$m$-Nearly $k$-Universal Words - Investigating Simon Congruence*

Determining the index of the Simon congruence is a long outstanding open problem. Two words $u$ and $v$ are called Simon congruent if they have the same set of scattered factors, which are parts of the word in the correct order but not necessarily consecutive, e.g., oath is a scattered factor of logarithm. Following the idea of scattered factor $k$-universality, we investigate $m$-nearly $k$-universality, i.e., words where $m$ scattered factors of length $k$ are absent, w.r.t. Simon congruence. We present a full characterisation as well as the index of the congruence for $m = 1$, $2$, and $3$. Moreover, we present a characterisation of the universality of repetitions.

### Archives 2022

The talks of 2022 are available here.

### Archives 2021

The talks of 2021 are available here.

### Archives 2020

The talks of 2020 are available here.

### Lectures on combinatorics on words

Several starting lectures by Anna Frid are available here.