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I am a researcher at VTI - Swedish National Road and Transport Research Institute.
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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]
R. Brignall, and J. Sliačan Combinatorial
specifications for juxtapositions of permutation
classes. Electronic Journal of Combinatorics, 26(4). 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]O. Pikhurko, J. Sliačan, and K. Tyros. Strong Forms of stability from flag algebra calculations.
Journal of Combinatorial Theory, Series B, Vol. 135, 2019.
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]