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## R. Brignall, and J. Sliačan Combinatorial specifications for juxtapositions of permutation classes. *preprint*

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]
## K. Bringmann, B. Doerr, A. Neumann, and J. Sliačan. Online checkpointing with improved worst-case guarantees. *INFORMS Journal on Computing* 27(3): 478-490. 2015

In the online checkpointing problem, the task is to continuously maintain a set of $k$ checkpoints that allow to rewind an ongoing computation faster than by a full restart. The only operation allowed is to replace an old checkpoint by the current state. Our aim are checkpoint placement strategies that minimize rewinding cost, i.e., such that at all times $T$ when requested to rewind to some time $t \leq T$ the number of computation steps that need to be redone to get to $t$ from a checkpoint before $t$ is as small as possible. In particular, we want that the closest checkpoint earlier than $t$ is not further away from $t$ than $q_k$ times the ideal distance $T / (k+1)$, where $q_k$ is a small constant. Improving over earlier work showing $1 + 1/k \leq q_k \leq 2$, we show that $q_k$ can be chosen asymptotically less than $2$. We present algorithms with asymptotic discrepancy $q_k \leq 1.59 + o(1)$ valid for all k and $q_k \leq \ln(4) + o(1) \leq 1.39 + o(1)$ valid for k being a power of two. Experiments indicate the uniform bound $p_k \leq 1.7$ for all $k$. For small $k$, we show how to use a linear programming approach to compute good checkpointing algorithms. This gives discrepancies of less than $1.55$ for all $k < 60$. We prove the first lower bound that is asymptotically more than one, namely $q_k \geq 1.30 - o(1)$. We also show that optimal algorithms (yielding the infimum discrepancy) exist for all $k$. [arXiv:1302.4216]