Enumeration of Pin-Permutations

39 pagesInternational audienceIn this paper, we study the class of pin-permutations, that is to say of permutations having a pin representation. This class has been recently introduced in an article of Brignall, Huczinska and Vatter, where it is used to find properties (algebraicity of the generating function, decidability of membership) of classes of permutations, depending on the simple permutations this class contains. We give a recursive characterization of the substitution decomposition trees of pin-permutations, which allows us to compute the generating function of this class, and consequently to prove, as it is conjectured, the rationality of this generating function. Moreover, we show that the basis of the pin-permutation class is infinite

Enumeration of Stack-Sorting Preimages via a Decomposition Lemma

We give three applications of a recently-proven "Decomposition Lemma," which allows one to count preimages of certain sets of permutations under West's stack-sorting map $s$. We first enumerate the permutation class $s^{-1}(\text{Av}(231,321))=\text{Av}(2341,3241,45231)$, finding a new example of an unbalanced Wilf equivalence. This result is equivalent to the enumeration of permutations sortable by ${\bf B}\circ s$, where ${\bf B}$ is the bubble sort map. We then prove that the sets $s^{-1}(\text{Av}(231,312))$, $s^{-1}(\text{Av}(132,231))=\text{Av}(2341,1342,\underline{32}41,\underline{31}42)$, and $s^{-1}(\text{Av}(132,312))=\text{Av}(1342,3142,3412,34\underline{21})$ are counted by the so-called "Boolean-Catalan numbers," settling a conjecture of the current author and another conjecture of Hossain. This completes the enumerations of all sets of the form $s^{-1}(\text{Av}(\tau^{(1)},\ldots,\tau^{(r)}))$ for $\{\tau^{(1)},\ldots,\tau^{(r)}\}\subseteq S_3$ with the exception of the set $\{321\}$. We also find an explicit formula for $|s^{-1}(\text{Av}_{n,k}(231,312,321))|$, where $\text{Av}_{n,k}(231,312,321)$ is the set of permutations in $\text{Av}_n(231,312,321)$ with $k$ descents. This allows us to prove a conjectured identity involving Catalan numbers and order ideals in Young's lattice.Comment: 20 pages, 4 figures. arXiv admin note: text overlap with arXiv:1903.0913

Some statistics on permutations avoiding generalized patterns

In the last decade a huge amount of articles has been published studying pattern avoidance on permutations. From the point of view of enumeration, typically one tries to count permutations avoiding certain patterns according to their lengths. Here we tackle the problem of refining this enumeration by considering the statistics "first/last entry". We give complete results for every generalized patterns of type $(1,2)$ or $(2,1)$ as well as for some cases of permutations avoiding a pair of generalized patterns of the above types.Comment: 5 figure