Talk 1 Speaker: Sergey Kitaev (University of Strathclyde, UK) Title: Singleton mesh patterns in multidimensional permutations Date: 5:00-6:30 pm, Wednesday 31th May 2023 Venue: https://us02web.zoom.us/j/89540203417?pwd=WWlYekU2V1ozdEcvcEhXcjYxTXpXQT09 Abstract: Permutation patterns is a popular area of research introduced in 1968, but with roots going to the work of Leonhard Euler in 1749. In this talk, I will present a brand-new notion of a singleton mesh pattern (SMP), which is a multidimensional mesh pattern of length 1. It turns out that avoidance of this pattern in arbitrary large multi-dimensional permutations can be characterized using an invariant of a pattern called its rank. This allows to determine avoidability for an SMP P efficiently, even though determining rank of P is an NP-complete problem. Moreover, using the notion of a minus-antipodal pattern, one can characterize SMPs which occur at most once in any d-dimensional permutation. I will also discuss a number of enumerative results regarding the distributions of certain general projective, plus-antipodal, minus-antipodal and hyperplane SMPs. This is joint work with Sergey Avgustinovich, Jeffrey Liese, Vladimir Potapov and Anna Taranenko. Talk 2 Speaker: Sergey Kitaev (University of Strathclyde, UK) Title: Naturally labelled posets and a hierarchy related to interval orders Date: 5:00-6:30 pm, Thursday 1st June 2023 Venue: https://us02web.zoom.us/j/83805213689?pwd=YU9WeGpBdTNIcS9vY0JCZG1QcEVnZz09 Abstract: A partially ordered set (poset) (P,<_P) is naturally labelled by numbers in {1,2,...,n} if x <_P y implies x<y. Naturally labelled posets are in one-to-one correspondence with certain lower triangular binary matrices called poset matrices. Restricting naturally labelled posets (considering (2+2)-free, k-free, 3+1-free, N-free, etc, posets) we obtain combinatorial objects fitting nicely to a hierarchy related to interval orders that involves, for example, Fishburn matrices, factorial posets, ascent sequences, pattern-avoiding permutations, and many more objects. In particular, it turns out that (2+2,3)-free naturally labelled posets are in one-to-one correspondence with permutations avoiding the vincular pattern 12-34. In my presentation, I will introduce the objects in question, and will discuss the hierarchy along with open (embedding) problems. This is joint work with David Bevan and Gi-Sang Cheon