### Neetu Jangid: Asymptotics of Young diagrams through Matrix Models

In this talk, we will talk about a matrix model description of growth processes of Young diagrams. In particular, we will see that the Plancherel growth process and its generalisations can be described through unitary matrix models. The classical solutions of unitary matrix models capture the asymptotic behaviour of Young diagrams growing according to (q-deformed) Plancherel probability. We will also talk about a Hilbert space description of unitary matrix models and Young diagrams.

### Cian O'Brien: Alternating Sign Hypermatrices and Latin-like Squares

An *alternating sign matrix (ASM)* is a matrix with entries from
$\{-1,0,1\}$ for which the non-zero entries in each row and column
alternate in sign, starting and ending with $+1$. A *Latin
square* is an $n \times n$ array containing $n$ symbols such that
each row and column contains each symbol exactly once. A Latin square
$L$ can be interpreted as the weighted sum $L = \sum_{k=1}^n kP_k$ of
a unique sequence of $n \times n$ *mutually orthogonal*
permutation matrices $P_1, P_2, \dots, P_n$. As ASMs generalise
permutation matrices, Brualdi and Dahl (2018) used this interpretation
to generalise the concept of a Latin square by replacing the sequence
of permutation matrices with an *alternating sign
hypermatrix*. This talk presents some problems posed by Brualdi and
Dahl, some of which I have addressed in a previous publication, and
some of which are current work.

### Paolo Sentinelli: Non-commuting graphs and their universality

For any positive integer $n$, an $n$-universal graph is a graph which contains (as induced subgraphs) all the graphs on $n$ vertices. Considering a set of projections on quotients of the symmetric group $S_{n+1}$, I will define their non-commuting graph $G_n$, whose vertices are subsets of $\{1, 2, \dots \}$. The graph $G_n$ is proved to be $n$-universal for forests, and this is done by providing an explicit labeling for any forest on $n$ vertices. Moreover, I will state the conjecture that $G_n$ is $n$-universal. The results I will present have been published in the Europ. J. Combin. (2020).

### Christian Stump: Counting inversions and descents of random elements in finite Coxeter groups

Permutation statistics (this is, assigning numbers to permutations) is a fundamental concept in Enumerative Combinatorics. Among the most important are the Mahonian and Eulerian numbers given by the number of inversions and by the number of descents. In this talk I report on Mahonian and Eulerian statistics in general finite Coxeter groups by discussing properties of their probability distributions that we first exhibited using the Combinatorial Statistics Database. This talk is based on joint work with Thomas Kahle and recent generalizations jointly with Kathrin Meier.

### Bridget Eileen Tenner: Permutations, shapes, and insertion tableaux

Permutations $w \in S_n$ are famously in bijection with pairs of standard Young tableaux having shape λ(w), where $\lambda(w)$ is a partition of $n$. That bijection can be exploited to reveal structural connections between permutations and their shapes, and those relationships have important combinatorial implications. In this talk, we add to this literature, giving combinatorial meaning to the lengths of the rows below the top row, separate from the meaning given by Schensted's theorem, and we reveal a connection between a permutation's reduced words and the contents of its corresponding tableaux.