Heilbronn Institute for Mathematical Research

Abstract A factorisation problem in the symmetric group is central if conjugate permutations always have the same number of factorisations. We give the first fully combinatorial proof of the centrality of transitive star factorisations that is valid in all genera, which answers a natural question of Goulden and Jackson from 2009. We begin by showing that the set of star factorisations is equinumerous with a certain set of monotone factorisations, a new result. We give more than one proof of this, and, crucially, one of our proofs is bijective. As a corollary we obtain new formulae for some monotone double Hurwitz factorisations, and a new relation between Hurwitz and monotone Hurwitz factorisations. We also generalise a theorem of Goulden and Jackson from 2009 that states that the transitive power of Jucys–Murphy elements are central. Our theorem states that the transitive image of any symmetric function evaluated at Jucys–Murphy elements is central, which gives a transitive version of Jucys' original result from 1974.

Abstract In the graph shotgun assembly problem, we are given the balls of radius r around each vertex of a graph and asked to reconstruct the graph. We study the shotgun assembly of the Erdős-Rényi random graph $${\mathcal {G}}(n,p)$$ G ( n , p ) for a wide range of values of r. We determine the threshold for reconstructibility for each $$r\ge 3$$ r ≥ 3 , extending and improving substantially on results of Mossel and Ross for $$r=3$$ r = 3 . For $$r=2$$ r = 2 , we give upper and lower bounds that improve on results of Gaudio and Mossel by polynomial factors. We also give a sharpening of a result of Huang and Tikhomirov for $$r=1$$ r = 1 .

In this paper we provide an alternative solution to a result by Juh\'{a}sz that the twisted conjugacy problem for odd dihedral Artin groups is solvable, that is, groups with presentation $G(m) = \langle a,b \; | \; _{m}(a,b) = {}_{m}(b,a) \rangle$, where $m\geq 3$ is odd, and $_{m}(a,b)$ is the word $abab \dots$ of length $m$, is solvable. Our solution provides an implementable linear time algorithm, by considering an alternative group presentation to that of a torus knot group, and working with geodesic normal forms. An application of this result is that the conjugacy problem is solvable in extensions of odd dihedral Artin groups.Comment: Published in the journal of Groups, Complexity, Cryptology

For given positive integers \(k\) and \(n\), a family \(\mathcal{F}\) of subsets of \(\{1,\dots,n\}\) is \(k\)-antichain saturated if it does not contain an antichain of size \(k\), but adding any set to \(\mathcal{F}\) creates an antichain of size \(k\). We use sat\(^*(n, k)\) to denote the smallest size of such a family. For all \(k\) and sufficiently large \(n\), we determine the exact value of sat\(^*(n, k)\). Our result implies that sat\(^*(n, k)=n(k-1)-\Theta(k\log k)\), which confirms several conjectures on antichain saturation. Previously, exact values for sat\(^*(n,k)\) were only known for \(k\) up to \(6\).We also prove a strengthening of a result of Lehman-Ron which may be of independent interest. We show that given \(m\) disjoint chains \(C^1,\dots,C^m\) in the Boolean lattice, we can create \(m\) disjoint skipless chains that cover the elements from \(\cup_{i=1}^mC^i\) (where we call a chain skipless if any two consecutive elements differ in size by exactly one).Mathematics Subject Classifications: 06A07, 05D99Keywords: Skipless chains, poset saturation, antichain saturation, Boolean lattice

Given a finite poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced copy of $\mathcal P$. The induced saturation number of $\mathcal P$, denoted by $\text{sat}^*(n,\mathcal P)$, is the size of the smallest $\mathcal P$-saturated family with ground set $[n]$. In this paper we prove that the saturation number for any given poset grows at worst polynomially. More precisely, we show that $\text{sat}^*(n, \mathcal P)=O(n^c)$, where $c\leq|\mathcal{P}|^2/4+1$ is a constant depending on $\mathcal P$ only. We obtain this result by bounding the VC-dimension of our family.

In this short note, we prove equidistribution results regarding three families of three-colour partitions recently introduced by Schlosser and Zhou. To do so, we prove an asymptotic formula for the infinite product $F_{a,c}(\zeta ; {\rm e}^{-z}) := \prod_{n \geq 0} \big(1- \zeta {\rm e}^{-(a+cn)z}\big)$ ($a,c \in \mathbb{N}$ with $0<a\leq c$ and $\zeta$ a root of unity) when $z$ lies in certain sectors in the right half-plane, which may be useful in studying similar problems. As a corollary, we obtain the asymptotic behaviour of the three-colour partition families at hand.

Given a matrix $M = (a_{i,j})$ a square is a $2 \times 2$ submatrix with entries $a_{i,j}$, $a_{i, j+s}$, $a_{i+s, j}$, $a_{i+s, j +s}$ for some $s \geq 0$, and a zero-sum square is a square where the entries sum to $0$. Recently, Arévalo, Montejano and Roldán-Pensado proved that all large $n \times n$ $\{-1,1\}$-matrices $M$ with discrepancy $|\sum a_{i,j}| \leq n$ contain a zero-sum square unless they are split. We improve this bound by showing that all large $n \times n$ $\{-1,1\}$-matrices $M$ with discrepancy at most $n^2/4$ are either split or contain a zero-sum square. Since zero-sum square free matrices with discrepancy at most $n^2/2$ are already known, this bound is asymptotically optimal.

We show that every finite abelian group occurs as the group of rational points of an ordinary abelian variety over F 2 \mathbb {F}_2 , F 3 \mathbb {F}_3 and F 5 \mathbb {F}_5 . We produce partial results for abelian varieties over a general finite field F q \mathbb {F}_q . In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over F q \mathbb {F}_q when q q is large. Finally, we show that every finite cyclic group arises as the group of rational points of infinitely many simple abelian varieties over F 2 \mathbb {F}_2 .

Abstract Detecting changepoints in data sets with many variates is a data science challenge of increasing importance. Motivated by the problem of detecting changes in the incidence of terrorism from a global terrorism database, we propose a novel approach to multiple changepoint detection in multivariate time series. Our method, which we call SUBSET, is a model-based approach which uses a penalised likelihood to detect changes for a wide class of parametric settings. We provide theory that guides the choice of penalties to use for SUBSET, and that shows it has high power to detect changes regardless of whether only a few variates or many variates change. Empirical results show that SUBSET out-performs many existing approaches for detecting changes in mean in Gaussian data; additionally, unlike these alternative methods, it can be easily extended to non-Gaussian settings such as are appropriate for modelling counts of terrorist events.