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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. 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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3. 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)
4. Composition-dependent shuffle-shear coupling and shuffle-regulated strain glass transition in compositionally modulated Ti-Nb alloys

   https://doi.org/10.1016/j.actamat.2023.118697  

   Su, Yunting; Liang, Chuanxin; Sun, Xun; Zhang, Hualei; Liang, Qianglong; Zheng, Yufeng; Hao, Yulin; Yang, Rui; Wang, Dong; Banerjee, Dipankar; et al (March 2023, Acta Materialia)
5. Difference operators via GKLO-type homomorphisms: shuffle approach and application to quantum Q-systems

   https://doi.org/10.1007/s11005-023-01639-1  

   Tsymbaliuk, Alexander (February 2023, Letters in Mathematical Physics)