On the effective and automatic enumeration of polynomial permutation classes

We describe an algorithm, implemented in Python, which can enumerate any permutation class with polynomial enumeration from a structural description of the class. In particular, this allows us to find formulas for the number of permutations of length n which can be obtained by a finite number of block sorting operations (e.g., reversals, block transpositions, cut-and-paste moves)

Pattern-Avoiding Involutions: Exact and Asymptotic Enumeration

We consider the enumeration of pattern-avoiding involutions, focusing in particular on sets defined by avoiding a single pattern of length 4. As we demonstrate, the numerical data for these problems demonstrates some surprising behavior. This strange behavior even provides some very unexpected data related to the number of 1324-avoiding permutations

Generating Permutations with Restricted Containers

We investigate a generalization of stacks that we call $\mathcal{C}$-machines. We show how this viewpoint rapidly leads to functional equations for the classes of permutations that $\mathcal{C}$-machines generate, and how these systems of functional equations can frequently be solved by either the kernel method or, much more easily, by guessing and checking. General results about the rationality, algebraicity, and the existence of Wilfian formulas for some classes generated by $\mathcal{C}$-machines are given. We also draw attention to some relatively small permutation classes which, although we can generate thousands of terms of their enumerations, seem to not have D-finite generating functions

The minimum Manhattan distance and minimum jump of permutations

Let $\pi$ be a permutation of $\{1,2,\ldots,n\}$. If we identify a permutation with its graph, namely the set of $n$ dots at positions $(i,\pi(i))$, it is natural to consider the minimum $L^1$ (Manhattan) distance, $d(\pi)$, between any pair of dots. The paper computes the expected value (and higher moments) of $d(\pi)$ when $n\rightarrow\infty$ and $\pi$ is chosen uniformly, and settles a conjecture of Bevan, Homberger and Tenner (motivated by permutation patterns), showing that when $d$ is fixed and $n\rightarrow\infty$, the probability that $d(\pi)\geq d+2$ tends to $e^{-d^2 - d}$. The minimum jump $mj(\pi)$ of $\pi$, defined by $mj(\pi)=\min_{1\leq i\leq n-1} |\pi(i+1)-\pi(i)|$, is another natural measure in this context. The paper computes the asymptotic moments of $mj(\pi)$, and the asymptotic probability that $mj(\pi)\geq d+1$ for any constant $d$.Comment: 20 pages, 3 figures. Minor changes throughou