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1. Cyclic Shuffle-Compatibility Via Cyclic Shuffle Algebras

   https://doi.org/10.1007/s00026-023-00669-9  

   Liang, Jinting; Sagan, Bruce_E; Zhuang, Yan (October 2023, Annals of Combinatorics)

   Abstract A permutation statistic$${{\,\textrm{st}\,}}$$ $\text{st}$is said to be shuffle-compatible if the distribution of$${{\,\textrm{st}\,}}$$ $\text{st}$over the set of shuffles of two disjoint permutations$$\pi $$ $\pi$and$$\sigma $$ $\sigma$depends only on$${{\,\textrm{st}\,}}\pi $$ $\text{st}\pi$,$${{\,\textrm{st}\,}}\sigma $$ $\text{st}\sigma$, and the lengths of$$\pi $$ $\pi$and$$\sigma $$ $\sigma$. Shuffle-compatibility is implicit in Stanley’s early work onP-partitions, and was first explicitly studied by Gessel and Zhuang, who developed an algebraic framework for shuffle-compatibility centered around their notion of the shuffle algebra of a shuffle-compatible statistic. For a family of statistics called descent statistics, these shuffle algebras are isomorphic to quotients of the algebra of quasisymmetric functions. Recently, Domagalski, Liang, Minnich, Sagan, Schmidt, and Sietsema defined a version of shuffle-compatibility for statistics on cyclic permutations, and studied cyclic shuffle-compatibility through purely combinatorial means. In this paper, we define the cyclic shuffle algebra of a cyclic shuffle-compatible statistic, and develop an algebraic framework for cyclic shuffle-compatibility in which the role of quasisymmetric functions is replaced by the cyclic quasisymmetric functions recently introduced by Adin, Gessel, Reiner, and Roichman. We use our theory to provide explicit descriptions for the cyclic shuffle algebras of various cyclic permutation statistics, which in turn gives algebraic proofs for their cyclic shuffle-compatibility. 

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2. To Shuffle or Not To Shuffle: Mini-Batch Shuffling Strategies for Multi-class Imbalanced Classification

   https://doi.org/10.1109/CSCI58124.2022.00057  

   Mao, Yuwei; Gupta, Vishu; Wang, Kewei; Liao, Wei-keng; Choudhary, Alok; Agrawal, Ankit (December 2022, 2022 International Conference on Computational Science and Computational Intelligence (CSCI))
3. Shuffle-Rational Series: Recognizability and Realizations

   https://doi.org/10.1109/ICSTCC50638.2020.9259634  

   Venkatesh, G. S.; Gray, W. Steven (October 2020, Proc. 24st International Conference on System Theory, Control and Computing)

   null (Ed.)

   The notion of a shuffle-rational formal power series is introduced. Then two equivalent characterizations are presented, one in terms of an analogue of Schützenberger’s recognizability of a series, and the other in the context of state space realizations of nonlinear input-output systems represented by Chen-Fliess series. An underlying computational framework is also described which employs the Hopf algebra associated with the shuffle group. As an application, it is shown how to model a bilinear system with output saturation in this context. 

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4. Deterministic Encryption with the Thorp Shuffle

   https://doi.org/10.1007/s00145-017-9262-z  

   Morris, Ben; Rogaway, Phillip; Stegers, Till (April 2018, Journal of Cryptology)
5. A Shuffle Theorem for Paths Under Any Line

   https://doi.org/10.1017/fmp.2023.4  

   Blasiak, Jonah; Haiman, Mark; Morse, Jennifer; Pun, Anna; Seelinger, George H. (January 2023, Forum of Mathematics, Pi)

   Abstract We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose x and y intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of $$\operatorname {\mathrm {GL}}_{l}$$ characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials. 

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