arXiv subjects

Jonas Sjöstrand

Publications and source records attributed to Jonas Sjöstrand.

14 recordsLinked to original sources

arXiv2023

Orbits under Dual Symplectic Transvections

Consider an arbitrary field $K$ and a finite-dimensional vector space $X$ over $K$ equipped with a, possibly degenerate, symplectic form $ω$. Given a spanning subset $S$ of $X$, for each $k$ in $K$ and each vector $s$ in $S$, consider the symplectic transvection mapping a vector $x$ to $x+kω(x,s)s$. The group generated by these transvections has been extensively studied, and its orbit structure is known. In this paper, we obtain corresponding results for the orbits of the dual action on $X^\ast$. As for the non-dual case, the analysis gets harder when the field contains only two elements. For that field, the dual transvection group is equivalent to a game known as the lit-only sigma game, played on a graph. Our results provide a complete solution to the reachability problem of that game, previously solved only for some special cases.

math.RT↗

arXiv2023

Continuity of Limit Surfaces of Locally Uniform Random Permutations

A locally uniform random permutation is generated by sampling $n$ points independently from some absolutely continuous distribution $ρ$ on the plane and interpreting them as a permutation by the rule that $i$ maps to $j$ if the $i$th point from the left is the $j$th point from below. As $n$ tends to infinity, decreasing subsequences in the permutation will appear as curves in the plane, and by interpreting these as level curves, a union of decreasing subsequences gives rise to a surface. In a recent paper by the author it was shown that, for any $r\ge0$, under the correct scaling as $n$ tends to infinity, the surface of the largest union of $\lfloor r\sqrt{n}\rfloor$ decreasing subsequences approaches a limit in the sense that it will come close to a maximizer of a specific variational integral (and, under reasonable assumptions, that the maximizer is essentially unique). In the present paper we show that there exists a continuous maximizer, provided that $ρ$ has bounded density and support. The key ingredient in the proof is a new theorem about real functions of two variables that are increasing in both variables: We show that, for any constant $C$, any such function can be made continuous without increasing the diameter of its image or decreasing anywhere the product of its partial derivatives clipped by $C$, that is the minimum of the product and $C$.

math.PR↗

arXiv2023

Monotone Subsequences in Locally Uniform Random Permutations

A locally uniform random permutation is generated by sampling $n$ points independently from some absolutely continuous distribution $ρ$ on the plane and interpreting them as a permutation by the rule that $i$ maps to $j$ if the $i$th point from the left is the $j$th point from below. As $n$ tends to infinity, decreasing subsequences in the permutation will appear as curves in the plane, and by interpreting these as level curves, a union of decreasing subsequences give rise to a surface. We show that, under the correct scaling, for any $r\ge0$, the largest union of $\lfloor r\sqrt{n}\rfloor$ decreasing subsequences approaches a limit surface as $n$ tends to infinity, and the limit surface is a solution to a specific variational problem. As a corollary, we prove the existence of a limit shape for the Young diagram associated to the random permutation under the Robinson-Schensted correspondence. In the special case where $ρ$ is the uniform distribution on the diamond $|x|+|y|<1$ we conjecture that the limit shape is triangular, and assuming the conjecture is true we find an explicit formula for the limit surfaces of a uniformly random permutation and recover the famous limit shape of Vershik, Kerov and Logan, Shepp.

math.PR↗

arXiv2022

Making multigraphs simple by a sequence of double edge swaps

We show that any loopy multigraph with a graphical degree sequence can be transformed into a simple graph by a finite sequence of double edge swaps with each swap involving at least one loop or multiple edge. Our result answers a question of Janson motivated by random graph theory, and it adds to the rich literature on reachability of double edge swaps with applications in Markov chain Monte Carlo sampling from the uniform distribution of graphs with prescribed degrees.

math.CO↗

arXiv2019

Learning to Personalize in Appearance-Based Gaze Tracking

Personal variations severely limit the performance of appearance-based gaze tracking. Adapting to these variations using standard neural network model adaptation methods is difficult. The problems range from overfitting, due to small amounts of training data, to underfitting, due to restrictive model architectures. We tackle these problems by introducing the SPatial Adaptive GaZe Estimator (SPAZE). By modeling personal variations as a low-dimensional latent parameter space, SPAZE provides just enough adaptability to capture the range of personal variations without being prone to overfitting. Calibrating SPAZE for a new person reduces to solving a small optimization problem. SPAZE achieves an error of 2.70 degrees with 9 calibration samples on MPIIGaze, improving on the state-of-the-art by 14 %. We contribute to gaze tracking research by empirically showing that personal variations are well-modeled as a 3-dimensional latent parameter space for each eye. We show that this low-dimensionality is expected by examining model-based approaches to gaze tracking. We also show that accurate head pose-free gaze tracking is possible.

cs.CV↗

arXiv2017

Limit shapes of stable configurations of a generalized Bulgarian solitaire

Bulgarian solitaire is played on $n$ cards divided into several piles; a move consists of picking one card from each pile to form a new pile. In a recent generalization, $σ$-Bulgarian solitaire, the number of cards you pick from a pile is some function $σ$ of the pile size, such that you pick $σ(h)\le h$ cards from a pile of size $h$. Here we consider a special class of such functions. Let us call $σ$ well-behaved if $σ(1)=1$ and if both $σ(h)$ and $h-σ(h)$ are non-decreasing functions of $h$. Well-behaved $σ$-Bulgarian solitaire has a geometric interpretation in terms of layers at certain levels being picked in each move. It also satisfies that if a stable configuration of $n$ cards exists it is unique. Moreover, if piles are sorted in order of decreasing size ($λ_1 \ge λ_2\ge \dots$) then a configuration is convex if and only if it is a stable configuration of some well-behaved $σ$-Bulgarian solitaire. If sorted configurations are represented by Young diagrams and scaled down to have unit height and unit area, the stable configurations corresponding to an infinite sequence of well-behaved functions ($σ_1, σ_2, \dots$) may tend to a limit shape $ϕ$. We show that every convex $ϕ$ with certain properties can arise as the limit shape of some sequence of well-behaved $σ_n$. For the special case when $σ_n(h)=\lceil q_n h \rceil$ for $0 < q_n \le 1$, these limit shapes are triangular (in case $q_n^2 n\rightarrow 0$), or exponential (in case $q_n^2 n\rightarrow \infty$), or interpolating between these shapes (in case $q_n^2 n\rightarrow C>0$).

math.CO↗

arXiv2015

The Limit Shape of a Stochastic Bulgarian Solitaire

We consider a stochastic version of Bulgarian solitaire: A number of cards are distributed in piles; in every round a new pile is formed by cards from the old piles, and each card is picked independently with a fixed probability. This game corresponds to a multi-square birth-and-death process on Young diagrams of integer partitions. We prove that this process converges in a strong sense to an exponential limit shape as the number of cards tends to infinity. Furthermore, we bound the probability of deviation from the limit shape and relate this to the number of rounds played in the solitaire.

math.PR↗

arXiv2015

Markov chains on graded posets: Compatibility of up-directed and down-directed transition probabilities

We consider two types of discrete-time Markov chains where the state space is a graded poset and the transitions are taken along the covering relations in the poset. The first type of Markov chain goes only in one direction, either up or down in the poset (an \emph{up chain} or \emph{down chain}). The second type toggles between two adjacent rank levels (an \emph{up-and-down chain}). We introduce two compatibility concepts between the up-directed transition probabilities (an \emph{up rule}) and the down-directed (a \emph{down rule}), and we relate these to compatibility between up-and-down chains. This framework is used to prove a conjecture about a limit shape for a process on Young's lattice. Finally, we settle the questions whether the reverse of an up chain is a down chain for some down rule and whether there exists an up or down chain at all if the rank function is not bounded.

math.PR↗

arXiv2014

Pomax games - a family of partizan games played on posets

We introduce the following class of partizan games, called pomax games. Given a partially ordered set whose elements are colored black or white, the players Black and White take turns removing any maximal element of their own color. If there is no such element, the player loses. We prove that pomax games are always integer-valued and for colored tree posets and chess-colored Young diagram posets we give a simple formula for the value of the game. However, for pomax games on general posets of height 3 we show that the problem of deciding the winner is PSPACE-complete and for posets of height 2 we prove NP-hardness. Pomax games are just a special case of a larger class of integer-valued games that we call element-removal games, and we pose some open questions regarding element-removal games that are not pomax games.

math.CO↗

arXiv2013

A product formula for the TASEP on a ring

For a random permutation sampled from the stationary distribution of the TASEP on a ring, we show that, conditioned on the event that the first entries are strictly larger than the last entries, the order of the first entries is independent of the order of the last entries. The proof uses multi-line queues as defined by Ferrari and Martin, and the theorem has an enumerative combinatorial interpretation in that setting. Finally, we present a conjecture for the case where the small and large entries are not separated.

math.PR↗

arXiv2011

Equidistributed statistics on matchings and permutations

We show that the bistatistic of right nestings and right crossings in matchings without left nestings is equidistributed with the number of occurrences of two certain patterns in permutations, and furthermore that this equidistribution holds when refined to positions of these statistics in matchings and permutations. For this distribution we obtain a non-commutative generating function which specializes to Zagier's generating function for the Fishburn numbers after abelianization. As a special case we obtain proofs of two conjectures of Claesson and Linusson. Finally, we conjecture that our results can be generalized to involving left crossings of matchings too.

math.CO↗

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↗

arXiv2005

On the sign-imbalance of partition shapes

Let the sign of a standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. A conjecture by Richard Stanley says that the sum of the signs of all SYTs with n squares is 2^[n/2]. We present a stronger theorem with a purely combinatorial proof using the Robinson-Schensted correspondence and a new concept called chess tableaux. We also prove a sharpening of another conjecture by Stanley concerning weighted sums of squares of sign-imbalances. The proof is built on a remarkably simple relation between the sign of a permutation and the signs of its RS-corresponding tableaux.

math.CO↗

arXiv2004

Bier spheres and posets

In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n-2)-spheres on 2n vertices as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual. Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that "cut across an ideal." Thus we arrive at a substantial generalization of Bier's construction: the Bier posets Bier(P,I) of an arbitrary bounded poset P of finite length. In the case of face posets of PL spheres this yields cellular "generalized Bier spheres." In the case of Eulerian or Cohen-Macaulay posets P we show that the Bier posets Bier(P,I) inherit these properties. In the boolean case originally considered by Bier, we show that all the spheres produced by his construction are shellable, which yields "many shellable spheres", most of which lack convex realization. Finally, we present simple explicit formulas for the g-vectors of these simplicial spheres and verify that they satisfy a strong form of the g-conjecture for spheres.

math.CO↗