Math Colloquium

### Speaker: Thomas Selig, University of Strathclyde

### Title: EW-tableaux, permutations and recurrent configurations of the sandpile model on Ferrers graphs.

Location: VRII, V-147

Time: Wednesday 27 June at 10:30

### Abstract:

The Abelian sandpile model (ASM) is a dynamic process on a graph. More specifically, it is a Markov chain on the set of configurations on that graph. Of particular interest are the recurrent configurations, i.e. those that appear infinitely often in the long-time running of the model. We study the ASM on Ferrers graphs, a class of bipartite graphs in one-to-one correspondence with Ferrers diagrams. We show that minimal recurrent configurations are in one-to-one correspondence with a set of certain 0/1 fillings of the Ferrers diagrams introduced by Ehrenborg and van Willigensburg. We refer to these fillings as EW-tableaux, and establish a bijection between the set of EW-tableaux of a given Ferrers diagram and a set of permutations whose descent bottoms are given by the shape of the Ferrers diagram. This induces a bijection between these permutations and minimal recurrent configurations of the ASM. We enrich this bijection to encode all recurrent configurations, via a decoration of the corresponding permutation. We also show that the set of recurrent configurations over all Ferrers graphs of a given size are in bijection with the set of alternating trees of that size.

Math Colloquium

### Speaker: Anders Claesson

Title: Interval orders via combinatorial species and ballot matrices

Location: V-157, VRII.

Time: Friday, Februar 5 at 13:20.

### Abstract:

We give a brief introduction to (some aspects of) combinatorial species.

Using this framework we introduce ballot matrices and present a subset

of them that is in bijection with labeled interval orders. Such ballot

matrices decompose naturally into a pair of permutations with related

properties, which leads to a new formula for the number of labeled

interval orders.

This talk is based on joint work with Stuart Hannah.

Math Colloquium

### Speaker: Bjarni Jens Kristinsson, University of Iceland, and Henning Úlfarsson, Reykjavik University

Title: Occurrence graphs of patterns in permutations

Location: Naustið, Endurmenntun (here)

Time: Thursday, June 04, at 15:00-16:00.

### Abstract:

This paper is based on a generalisation of the idea behind the proof of the Simultaneous Shading Lemma by Claesson et al. (2014). We define the occurrence graph \(G_p(\pi)\) of a pattern \(p\) in a permutation \(\pi\) as the graph with the occurrences of \(p\) in \(\pi\) as vertices and edges between the vertices if the occurrences differ by exactly one element. We study the general properties of the occurrence graphs and some interesting extreme cases. The main theorem in this paper is that every hereditary property of graphs produces a permutation class.

Math Colloquium

### Speaker: Henning Ulfarsson, Reykjavik University

Title: Struct: An algorithm for guessing the structure and enumeration of permutation sets

Location: Naustið, Endurmenntun (here)

Time: Tuesday January 22.,15:00-16:00.

### Authors:

Michael Albert, Anders Claesson, Bjarki Gudmundsson, Henning Ulfarsson

### Abstract:

Struct is an algorithm being developed by the authors to

guess the structure of a set of permutations. In some cases the structure

discovered is sufficient to infer the generating function of the set and

provides an enumeration of the permutations by length. A preliminary version of

the algorithm will be presented and applied to several sets of permutations.

This research is funded by the Icelandic Research Fund, Grant no.~141761-051