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Register a new userThe Foata correspondence, cycle lengths and anomalies
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Sanjay Ramassamy
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In their study of the densest jammed configurations for theater models, Krapivsky and Luck observe that two classes of permutations have the same cardinalities and ask for a bijection between them. In this note we show that the Foata correspondence provides the desired bijection.
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Woodall proved that for a graph $G$ of order $ngeq 2k+3$ where $kgeq 0$ is an integer, if $e(G)geq binom{n-k-1}{2}+binom{k+2}{2}+1$ then $G$ contains a $C_{ell}$ for each $ellin [3,n-k]$. In this article, we prove a stability result of this theorem. As a byproduct, we give complete solutions to two problems in cite{GN19}. Our second part is devoted to an open problem by Nikiforov: what is the maximum $C$ such that for all positive $varepsilon<C$ and sufficiently large $n$, every graph $G$ of order $n$ with spectral radius $rho(G)>sqrt{lfloorfrac{n^2}{4}rfloor}$ contains a cycle of length $ell$ for every $ellleq (C-varepsilon)n$. We prove that $Cgeqfrac{1}{4}$ by a method different from previous ones, improving the existing bounds. We also derive an ErdH{o}s-Gallai type edge number condition for even cycles, which may be of independent interest.
Visual semantic correspondence is an important topic in computer vision and could help machine understand objects in our daily life. However, most previous methods directly train on correspondences in 2D images, which is end-to-end but loses plenty of information in 3D spaces. In this paper, we propose a new method on predicting semantic correspondences by leveraging it to 3D domain and then project corresponding 3D models back to 2D domain, with their semantic labels. Our method leverages the advantages in 3D vision and can explicitly reason about objects self-occlusion and visibility. We show that our method gives comparative and even superior results on standard semantic benchmarks. We also conduct thorough and detailed experiments to analyze our network components. The code and experiments are publicly available at https://github.com/qq456cvb/SemanticTransfer.
Image co-segmentation is an active computer vision task that aims to segment the common objects from a set of images. Recently, researchers design various learning-based algorithms to undertake the co-segmentation task. The main difficulty in this task is how to effectively transfer information between images to make conditional predictions. In this paper, we present CycleSegNet, a novel framework for the co-segmentation task. Our network design has two key components: a region correspondence module which is the basic operation for exchanging information between local image regions, and a cycle refinement module, which utilizes ConvLSTMs to progressively update image representations and exchange information in a cycle and iterative manner. Extensive experiments demonstrate that our proposed method significantly outperforms the state-of-the-art methods on four popular benchmark datasets -- PASCAL VOC dataset, MSRC dataset, Internet dataset, and iCoseg dataset, by 2.6%, 7.7%, 2.2%, and 2.9%, respectively.
A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al., where 25 out of 65 non-equivalent cases were solved. In this paper, we give 27 distribution results for these patterns including 14 distributions for which avoidance was not known. Moreover, for the unsolved cases, we prove an equidistribution result (out of 6 equidistribution results we prove in total), and conjecture 6 more equidistributions. Finally, we find seemingly unknown distribution of the well known permutation statistic ``strict fixed point, which plays a key role in many of our enumerative results. This paper is the first systematic study of distributions of mesh patterns. Our techniques to obtain the results include, but are not limited to, obtaining functional relations for generating functions, and finding recurrence relations and bijections.
Necessary and sufficient conditions are derived for a social choice correspondence to be the one that selects the Pareto optimal alternatives.
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