arXiv subjects

Oscar A. Lopez

Publications and source records attributed to Oscar A. Lopez.

2 recordsLinked to original sources

arXiv2026

On separable permutations and three other pairs in the Schröder class

We study positional statistics for four families of pattern-avoiding permutations counted by the large Schröder numbers. Specifically, we focus on the pairs of patterns $\{2413,3142\}$ (separable permutations), $\{1324,1423\}$, $\{1423,2413\}$, and $\{1324,2134\}$. For each class, we derive multivariate generating functions that track the relative positions of specific entries. Our approach combines structural decompositions with the kernel method to obtain explicit formulas involving the generating function for the Schröder numbers. As a byproduct, we obtain alternative proofs that each of these classes is enumerated by the Schröder numbers. We also identify several known triangular arrays arising from our positional refinements, including connections to the central binomial coefficients and sequences appearing in the work of Kreweras on covering hierarchies.

math.CO↗

arXiv2024

A positional statistic for 1324-avoiding permutations

We consider the class $S_n(1324)$ of permutations of size $n$ that avoid the pattern 1324 and examine the subset $S_n^{a\prec n}(1324)$ of elements for which $a\prec n\prec [a-1]$, $a\ge 1$. This notation means that, when written in one line notation, such a permutation must have $a$ to the left of $n$, and the elements of $\{1,\dots,a-1\}$ must all be to the right of $n$. For $n\ge 2$, we establish a connection between the subset of permutations in $S_n^{1\prec n}(1324)$ having the 1 adjacent to the $n$ (called primitives), and the set of 1324-avoiding dominoes with $n-2$ points. For $a\in\{1,2\}$, we introduce constructive algorithms and give formulas for the enumeration of $S_n^{a\prec n}(1324)$ by the position of $a$ relative to the position of $n$. For $a\ge 3$, we formulate some conjectures for the corresponding generating functions.

math.CO↗