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   Christian Gaetz ; Sergi Elizalde ; Colin Defant and Nathan Williams ; Nicholas J. Williams ; Wenjie Fang ; Jesús A. De Loera and William J. Wesley ; Yasuhide Numata, Yusuke Takahashi ; Margaret Bayer, Mark Denker ; Christian Gaetz, Oliver Pechenik ; Jia Huang and Erkko Lehtonen ; et al ( April 2023 , Séminaire lotharingien de combinatoire)

   Krattenthaler, Christian ; Thibon, Jean-Yves (Ed.)
2. Learning mixtures of permutations: Groups of pairwise comparisons and combinatorial method of moments

   https://doi.org/10.1214/22-AOS2185  

   Mao, Cheng ; Wu, Yihong ( August 2022 , The Annals of Statistics)
3. Compute Choice: Learning Distributions over Permutations

   Shah, Devavrat ( October 2019 , Cambridge University Press bulletin)

   We discuss the question of learning distribution over permutations of a given set of choices, options or items based on partial observations. This is central to capturing the so called ``choice'' in a variety of contexts: understanding preferences of consumers over a collection of products based on purchasing and browsing data in the setting of retail and e-commerce, learning public opinion amongst a collection of socio-economic issues based on sparse polling data, deciding a ranking of teams or players based on outcomes of games, electing leaders based on votes, and more generally collaborative decision making based on collective judgement such as accepting paper(s) in a competitive academic conference. The question of learning distribution over permutations arises beyond capturing ``choice'' as well. For example, tracking a collection of objects using noisy cameras, or aggregating ranking of web-pages using outcomes of multiple search engines. It is only natural that such a topic has been extensively studied in Economics, Political Science and Psychology for more than a century, and more so recently in Computer Science, Electrical Engineering, Statistics and Operations Research. Here we shall focus on the task of learning distribution over permutations from its marginal distributions of two types: first-order marginals and pair-wise comparisons. There has been a lot of progress made on this topic in the last decade. The ideal goal is to provide a comprehensive overview of the state-of-art on this topic. We shall provide detailed overview of selective aspects, biased by author's perspective of the topic. And provide sufficient pointers to aspects not covered here. We shall emphasize on ability to identify the entire distribution over permutation as well as ``best ranking''. 

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4. Learning sparse mixtures of permutations from noisy information

   De, Anindya ; O'Donnell, Ryan ; Servedio, Rocco ( August 2021 , Proceedings of Machine Learning Research)
5. A Bijection Between Evil-Avoiding and Rectangular Permutations

   https://doi.org/10.37236/11841  

   Tung, Katherine ( October 2023 , The Electronic Journal of Combinatorics)

   Evil-avoiding permutations, introduced by Kim and Williams in 2022, arise in the study of the inhomogeneous totally asymmetric simple exclusion process. Rectangular permutations, introduced by Chirivì, Fang, and Fourier in 2021, arise in the study of Schubert varieties and Demazure modules. Taking a suggestion of Kim and Williams, we supply an explicit bijection between evil-avoiding and rectangular permutations in $S_n$ that preserves the number of recoils. We encode these classes of permutations as regular languages and construct a length-preserving bijection between words in these regular languages. We extend the bijection to another Wilf-equivalent class of permutations, namely the $1$-almost-increasing permutations, and exhibit a bijection between rectangular permutations and walks of length $2n-2$ in a path of seven vertices starting and ending at the middle vertex.

    

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