arXiv subjects

Svante Linusson

Publications and source records attributed to Svante Linusson.

At least 37 records · Page 2Linked to original sources

arXiv2018

Bumping sequences and multispecies juggling

Building on previous work by four of us (ABCN), we consider further generalizations of Warrington's juggling Markov chains. We first introduce "multispecies" juggling, which consist in having balls of different weights: when a ball is thrown it can possibly bump into a lighter ball that is then sent to a higher position, where it can in turn bump an even lighter ball, etc. We both study the case where the number of balls of each species is conserved and the case where the juggler sends back a ball of the species of its choice. In this latter case, we actually discuss three models: add-drop, annihilation and overwriting. The first two are generalisations of models presented in (ABCN) while the third one is new and its Markov chain has the ultra fast convergence property. We finally consider the case of several jugglers exchanging balls. In all models, we give explicit product formulas for the stationary probability and closed form expressions for the normalisation factor if known.

math.CO↗

arXiv2017

Continuous Multi-line Queues and TASEP

In this paper, we study a distribution of labeled particles on a continuous ring. It arises in three different ways, all related to the multi-type TASEP on a ring. We prove formulas for the probability density function for some permutations and give conjectures for a larger class. We give a complete conjecture for the probability of two particles i, j being next to each other on the cycle, for which we prove some cases. We also find that two natural events associated to the process have exactly the same probability expressed as a Vandermonde determinant. It is unclear whether this is just a coincidence or a consequence of a deeper connection.

math.CO↗

arXiv2014

Correlations in the Multispecies TASEP and a Conjecture by Lam

We study correlations in the multispecies TASEP on a ring. Results on correlation of two adjacent points prove two conjectures by Thomas Lam on (a) the limiting direction of a reduced random walk in $\tilde A_{n-1}$ and (b) the asymptotic shape of a random integer partition with no hooks of length $n$, a so called $n$-core. We further investigate two-point correlations far apart and three-point nearest neighbour correlations and prove explicit formulas in almost all cases. These results can be seen as a finite strengthening of correlations in the TASEP speed process by Amir, Angel and Valkó. We also give conjectures for certain higher order nearest neighbour correlations. We find an unexplained independence property (provably for two points, conjecturally for more points) between points that are closer in position than in value that deserves more study.

math.PR↗

arXiv2014

An Inhomogeneous Multispecies TASEP on a Ring

We reinterpret and generalize conjectures of Lam and Williams as statements about the stationary distribution of a multispecies exclusion process on the ring. The central objects in our study are the multiline queues of Ferrari and Martin. We make some progress on some of the conjectures in different directions. First, we prove their conjectures in two special cases by generalizing the rates of the Ferrari-Martin transitions. Secondly, we define a new process on multiline queues, which have a certain minimality property. This gives another proof for one of the special cases; namely arbitrary jump rates for three species.

math.PR↗

arXiv2013

Dynamic adjustment: an electoral method for relaxed double proportionality

We describe an electoral system for distributing seats in a parliament. It gives proportionality for the political parties and close to proportionality for constituencies. The system suggested here is a version of the system used in Sweden and other Nordic countries with permanent seats in each constituency and adjustment seats to give proportionality on the national level. In the national election of 2010 the current Swedish system failed to give proportionality between parties. We examine here one possible cure for this unwanted behavior. The main difference compared to the current Swedish system is that the number of adjustment seats is not fixed, but rather dynamically determined to be as low as possible and still insure proportionality between parties.

math.OC↗

arXiv2013

First critical probability for a problem on random orientations in $G(n,p)$

We study the random graph $G(n,p)$ with a random orientation. For three fixed vertices $s,a,b$ in $G(n,p)$ we study the correlation of the events $a \to s$ and $s\to b$. We prove that asymptotically the correlation is negative for small $p$, $p<\frac{C_1}n$, where $C_1\approx0.3617$, positive for $\frac{C_1}n<p<\frac2n$ and up to $p=p_2(n)$. Computer aided computations suggest that $p_2(n)=\frac{C_2}n$, with $C_2\approx7.5$. We conjecture that the correlation then stays negative for $p$ up to the previously known zero at $\frac12$; for larger $p$ it is positive.

math.PR↗

arXiv2013

Correlation of paths between distinct vertices in a randomly oriented graph

We prove that in a random tournament the events $\{s\rightarrow a\}$ and $\{t\rightarrow b\}$ are positively correlated, for distinct vertices $a,s,b,t \in K_n.$ It is also proven that the correlation between the events $\{s\rightarrow a\}$ and $\{t\rightarrow b\}$ in the random graphs $G(n,p)$ and $G(n,m)$ with random orientation is positive for every fixed $p>0$ and sufficiently large $n$ (with $m=\left\lfloor p \binom{n}{2}\right\rfloor$). We conjecture it to be positive for all $p$ and all $n$. An exact recursion for $¶(\{s\rightarrow a\} \cap \{t\rightarrow b\})$ in $\gnp$ is given.

math.CO↗

arXiv2012

Rees products and lexicographic shellability

We use the theory of lexicographic shellability to provide various examples in which the rank of the homology of a Rees product of two partially ordered sets enumerates some set of combinatorial objects, perhaps according to some natural statistic on the set. Many of these examples generalize a result of J. Jonsson, which says that the rank of the unique nontrivial homology group of the Rees product of a truncated Boolean algebra of degree $n$ and a chain of length $n-1$ is the number of derangements in $§_n$.\

math.CO↗

arXiv2011

The probability of the Alabama paradox

Hamilton's method (also called method of largest remainder) is a natural and common method to distribute seats proportionally between states (or parties) in a parliament. In USA it has been abandoned due to some drawbacks, in particular the possibility of the Alabama paradox, but it is still in use in many other countries. In this paper we give, under certain assumptions, a closed formula for the asymptotic probability, as the number of seats tends to infinity, that the Alabama paradox occurs given the vector p_1,...,p_m of relative sizes of the states. From the theorem we deduce a number of consequences. For example it is shown that the expected number of states that will suffer from the Alabama paradox is asymptotically bounded above by 1/e. For random (uniformly distributed) relative sizes p_1,...,p_m the expected number of states to suffer from the Alabama paradox converges to slightly more than a third of this, or approximately 0.335/e=0.123, as m tends to infinity. We leave open the generalization of our formula to all possible (in particular rational) p_1,...,p_m.

math.PR↗

arXiv2010

Thomassen's Choosability Argument Revisited

Thomassen (1994) proved that every planar graph is 5-choosable. This result was generalised by {Š}krekovski (1998) and He et al. (2008), who proved that every $K_5$-minor-free graph is 5-choosable. Both proofs rely on the characterisation of $K_5$-minor-free graphs due to Wagner (1937). This paper proves the same result without using Wagner's structure theorem or even planar embeddings. Given that there is no structure theorem for graphs with no $K_6$-minor, we argue that this proof suggests a possible approach for attacking the Hadwiger Conjecture.

math.CO↗

arXiv2010

n! matchings, n! posets

We show that there are $n!$ matchings on $2n$ points without, so called, left (neighbor) nestings. We also define a set of naturally labeled $(2+2)$-free posets, and show that there are $n!$ such posets on $n$ elements. Our work was inspired by Bousquet-Mélou, Claesson, Dukes and Kitaev [J. Combin. Theory Ser. A. 117 (2010) 884--909]. They gave bijections between four classes of combinatorial objects: matchings with no neighbor nestings (due to Stoimenow), unlabeled $(2+2)$-free posets, permutations avoiding a specific pattern, and so called ascent sequences. We believe that certain statistics on our matchings and posets could generalize the work of Bousquet-Mélou et al.\ and we make a conjecture to that effect. We also identify natural subsets of matchings and posets that are equinumerous to the class of unlabeled $(2+2)$-free posets. We give bijections that show the equivalence of (neighbor) restrictions on nesting arcs with (neighbor) restrictions on crossing arcs. These bijections are thought to be of independent interest. One of the bijections maps via certain upper-triangular integer matrices that have recently been studied by Dukes and Parviainen [Electron. J. Combin. 17 (2010) \#R53]

math.CO↗

arXiv2010

Correlations for paths in random orientations of G(n,p) and G(n,m)

We study random graphs, both $G(n,p)$ and $G(n,m)$, with random orientations on the edges. For three fixed distinct vertices s,a,b we study the correlation, in the combined probability space, of the events a -> s and s -> b. For G(n,p), we prove that there is a p_c=1/2 such that for a fixed p p_c the correlation is positive for large enough n. We conjecture that for a fixed n\ge 27 the correlation changes sign three times for three critical values of p. For G(n,m) it is similarly proved that, with $p=m/\binom{n}{2}$, there is a critical p_c that is the solution to a certain equation and approximately equal to 0.7993. A lemma, which computes the probability of non existence of any k directed edges in G(n,m), is thought to be of independent interest. We present exact recursions to compute P(a -> s)$ and P(a -> s, s -> b)$. We also briefly discuss the corresponding question in the quenched version of the problem.

math.PR↗

arXiv2009

On percolation and the bunkbed conjecture

We study a problem on edge percolation on product graphs $G\times K_2$. Here $G$ is any finite graph and $K_2$ consists of two vertices $\{0,1\}$ connected by an edge. Every edge in $G\times K_2$ is present with probability $p$ independent of other edges. The Bunkbed conjecture states that for all $G$ and $p$ the probability that $(u,0)$ is in the same component as $(v,0)$ is greater than or equal to the probability that $(u,0)$ is in the same component as $(v,1)$ for every pair of vertices $u,v\in G$. We generalize this conjecture and formulate and prove similar statements for randomly directed graphs. The methods lead to a proof of the original conjecture for special classes of graphs $G$, in particular outerplanar graphs.

math.CO↗

arXiv2009

A counter-intuitive correlation in a random tournament

Consider a randomly oriented graph $G=(V,E)$ and let $a$, $s$ and $b$ be three distinct vertices in $V$. We study the correlation between the events $\{a\to s\}$ and $\{s\to b\}$. We show that, when $G$ is the complete graph $K_n$, the correlation is negative for $n=3$, zero for $n=4$, and that, counter-intuitively, it is positive for $n\ge 5$. We also show that the correlation is always negative when $G$ is a cycle, $C_n$, and negative or zero when $G$ is a tree (or a forest).

math.PR↗

arXiv2009

A note on correlations in randomly oriented graphs

Given a graph $G$, we consider the model where $G$ is given a random orientation by giving each edge a random direction. It is proven that for $a,b,s\in V(G)$, the events $\{s\to a\}$ and $\{s\to b\}$ are positively correlated. This correlation persists, perhaps unexpectedly, also if we first condition on $\{s\nto t\}$ for any vertex $t\neq s$. With this conditioning it is also true that $\{s\to b\}$ and $\{a\to t\}$ are negatively correlated. A concept of increasing events in random orientations is defined and a general inequality corresponding to Harris inequality is given. The results are obtained by combining a very useful lemma by Colin McDiarmid which relates random orientations with edge percolation, with results by van den Berg, Häggström, Kahn on correlation inequalities for edge percolation. The results are true also for another model of randomly directed graphs.

math.PR↗

arXiv2007

From Bruhat intervals to intersection lattices and a conjecture of Postnikov

We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation $w\in \Sn$ is at most the number of elements below $w$ in the Bruhat order, and (B) that equality holds if and only if $w$ avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups. A byproduct of this result and its proof is a set of inequalities relating Betti numbers of complexified inversion arrangements to Betti numbers of closed Schubert cells. Another consequence is a simple combinatorial interpretation of the chromatic polynomial of the inversion graph of a permutation which avoids the above patterns.

math.CO↗

arXiv2007

On the independence complex of square grids

The enumeration of independent sets of regular graphs is of interest in statistical mechanics, as it corresponds to the solution of hard-particle models. In 2004, it was conjectured by Fendleyet al. that for some rectangular grids, with toric boundary conditions, the alternating number of independent sets is extremely simple. More precisely, under a coprimality condition on the sides of the rectangle, the number of independent sets of even and odd cardinality always differ by 1. In physics terms, this means looking at the hard-particle model on these grids at activity -1. This conjecture was recently proved by Jonsson. Here we produce other families of grid graphs, with open or cylindric boundary conditions, for which similar properties hold without any size restriction: the number of independent sets of even and odd cardinality always differ by 0, 1,-1, or, in the cylindric case, by some power of 2. We show that these results reflect a stronger property of the independence complexes of our graphs. We determine the homotopy type of these complexes using Forman's discrete Morse theory. We find that these complexes are either contractible, or homotopic to a sphere, or, in the cylindric case, to a wedge of spheres. Finally, we use our enumerative results to determine the spectra of certain transfer matrices describing the hard-particle model on our graphs at activity -1. These results parallel certain conjectures of Fendley et al., proved by Jonsson in the toric case.

math.CO↗

arXiv2004

Complexes of graphs with bounded matching size

For positive integers k,n, we investigate the simplicial complex NM_k(n) of all graphs G on vertex set [n] such that every matching in G has size less than k. This complex (along with other associated cell complexes) is found to be homotopy equivalent to a wedge of spheres. The number and dimension of the spheres in the wedge are determined, and (partially conjectural) links to other combinatorially defined complexes are described. In addition we study for positive integers r,s and k the simplicial complex BNM_k(n) of all bipartite graphs G on bipartition [r] \cup [\bar{s}] such that there is no matching of size k in G, and obtain results similar to those obtained for NM_k(n).

math.CO↗