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The study of pattern containment and avoidance for linear permutations is a well-established area of enumerative combinatorics. A cyclic permutation is the set of all rotations of a linear permutation. Callan initiated the study of permutation avoidance in cyclic permutations and characterized the avoidance classes for all single permutations of length 4. We continue this work. In particular, we establish a cyclic variant of the Erdős-Szekeres Theorem that any linear permutation of length mn + 1 must contain either the increasing pattern of length m + 1 or the decreasing pattern of length n + 1. We then derive results about avoidance of multiple patterns of length 4. We also determine generating functions for the cyclic descent statistic on these classes. Finally, we end with various open questions and avenues for future research.

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Density functional theory (DFT) calculations were carried out in this work to systematically investigate the adsorption properties of monolayer black phosphorene (BP) for actinide uranium (U), thorium (Th), and plutonium (Pu) atoms. More specifically, the linear response method was used and the Hubbard U values of 2.97 and 2.61 eV were fitted describing the strong lattice point Coulomb interactions of the U and Pu 5f orbitals, respectively. From the DFT + U calculations, it was demonstrated that the U, Th, and Pu atoms can be favorably adsorbed on the hollow sites of the BP surface, with an adsorption energy of 3.48, 4.93, and 0.98 eV, respectively.By analyzing the electronic structure, charge transfer, and highest occupied molecular orbital, it was revealed that the U and Th adatoms can induce the generation of impurity states within the band gap of BP and stabilize their adsorption by strong p-d coupling with the phosphorus atoms. In addition, from the electron-phonon coupling calculations, it was revealed that the temperature for the stable adsorption of the U and Th adatoms can reach as high as 500 and 673 K. By using the adsorption rate equation, the critical temperature for the adsorption-desorption transition of U and Pu adatoms was estimated to be 632 and 185 K, respectively, while the Th adatom did not tend to desorb. Our results clearly indicate that BP is an ideal adsorption and separation material for actinides with promising potential applications, such as uranium extraction from seawater.