From Permutation Patterns to the Periodic Table

Abstract

(The above abstract has been extracted by the translator from the original article (L. Pudwell, From Permutation Patterns to the Periodic Table, Notices of the American Mathematical Society, 67 994–1001.))Abstract: Permutation patterns is a burgeoning area of research with roots in enumerative combinatorics and theoretical computer science. This article first presents a brief overview of pattern avoidance and a survey of enumeration results that are standard knowledge within the field. Then, we turn our attention to a newer optimization problem of pattern packing. We survey pattern packing results in the general case before we consider packing in a specific type of permutation that leads to a new and surprising connection with physical chemistry. Note that the original paper has published in ``Notices of the American Mathematical Society, 67, Number 7, 994-1001" and we have translated it into Farsi. This is just an extended abstract for Journal of Mathematics and Society. 1. IntroductionLet $S_k $ be the set of all permutations on $[k]=\{1, 2,\ldots,k\}$. Given $\pi \in S_k$ and $\rho \in S_l$, we say that $\pi$ contains $\rho$ as a pattern if there exist $1\leq i_1\pi_310?$ What other chemical or physical structures can be described in terms of pattern packing or pattern avoidance? Are there other combinatorial structures that give alternate ways to generate the sequences of atomic numbers of particular groups of chemical elements? The variety of applications of permutation patterns has grown tremendously in recent decades, and modeling electron orbitals can now be added to the list.

Keywords

• pattern avoidance
• permutation
• periodic table