## About

I currently work as a software quality engineer at Red Hat.

In 2022, I plan to ride Tour Te Waipounamu (23 Jan) and HT550 (28 May).

## Some output

(click on arrows to expand, on title to go to arxiv)

## R. Brignall, and J. Sliačan Combinatorial
specifications for juxtapositions of permutation
classes. *Electronic Journal of Combinatorics*, 26(4). 2019

We show that, given a suitable combinatorial specification for a permutation class $\mathcal{C}$, one can obtain a specification for the juxtaposition (on either side) of $\mathcal{C}$ with Av(21) or Av(12), and that if the enumeration for $\mathcal{C}$ is given by a rational or algebraic generating function, so is the enumeration for the juxtaposition. Furthermore this process can be iterated, thereby providing an effective method to enumerate any 'skinny' $k\times 1$ grid class in which at most one cell is non-monotone, with a guarantee on the nature of the enumeration given the nature of the enumeration of the non-monotone cell. [arXiv:1902.02705]

## O. Pikhurko, J. Sliačan, and K. Tyros. Strong Forms of stability from flag algebra calculations.
*Journal of Combinatorial Theory, Series B, Vol. 135, 2019.*

Given a hereditary family $\mathcal{G}$ of admissible graphs and a function $\lambda(G)$ that linearly depends on the statistics of order-$\kappa$ subgraphs in a graph $G$, we consider the extremal problem of determining $\lambda(n,\mathcal{G})$, the maximum of $\lambda(G)$ over all admissible graphs $G$ of order $n$. We call the problem perfectly $B$-stable for a graph $B$ if there is a constant $C$ such that every admissible graph $G$ of order $n\geq C$ can be made into a blow-up of $B$ by changing at most $C(\lambda(n,\mathcal{G})−\lambda(G))\binom{n}{2}$ adjacencies. As special cases, this property describes all almost extremal graphs of order $n$ within $o(n^2)$ edges and shows that every extremal graph of order $n\geq n_0$ is a blow-up of $B$.

We develop general methods for establishing stability-type results from flag algebra computations and apply them to concrete examples. In fact, one of our sufficient conditions for perfect stability is stated in a way that allows automatic verification by a computer. This gives a unifying way to obtain computer-assisted proofs of many new results. [arXiv:1706.02612]

## J. Sliačan, and W. Stromquist. Improving
bounds on packing densities of 4-point permutations. *DMTCS*, 19(2). 2018. Permutation Patterns 2016

We consolidate what is currently known about packing densities of 4-point permutations and in the process improve
current lower bounds for the packing densities of 1324 and 1342. We also provide rigorous upper bounds for the
packing densities of 1324, 1342, and 2413. All our bounds are within $10^{−4}$ of the true packing densities.
Together with the known bounds, this gives us a fairly complete picture of all 4-point packing densities. We also
provide new upper bounds for several small permutations of length at least five. Our main tool for the upper
bounds is the framework of flag algebras introduced by Razborov in 2007. [arXiv:1704.02959]

## R. Brignall, and J. Sliačan. Juxtaposing
Catalan permutation classes with monotone ones. *Electronic Journal of Combinatorics*, 24(2). 2017

This paper enumerates all juxtaposition classes of the form "$Av(abc)$ next to $Av(xy)$", where $abc$ is a
permutation of length three and $xy$ is a permutation of length two. We use Dyck paths decorated by sequences of
points to represent elements from such a juxtaposition class. Context-free grammars are then used to enumerate
these decorated Dyck paths. [arXiv:1302.4216]