# Semi-Baxter and strong-Baxter: two relatives of the Baxter sequence

## TL;DR

This paper introduces semi-Baxter and strong-Baxter permutations, enumerates them, proves conjectures, and derives their generating functions, revealing their combinatorial properties and asymptotic behaviors.

## Contribution

It defines two new permutation families avoiding specific vincular patterns, provides their enumeration, and establishes their generating functions and combinatorial structures.

## Key findings

- Semi-Baxter numbers enumerate plane permutations avoiding 2-14-3.
- The generating function for semi-Baxter permutations is D-finite and explicitly derived.
- Strong-Baxter permutations' generating function relates to quarter plane walks and is non D-finite.

## Abstract

In this paper, we enumerate two families of pattern-avoiding permutations: those avoiding the vincular pattern $2-41-3$, which we call semi-Baxter permutations, and those avoiding the vincular patterns $2-41-3$, $3-14-2$ and $3-41-2$, which we call strong-Baxter permutations. We call semi-Baxter numbers and strong-Baxter numbers the associated enumeration sequences. We prove that the semi-Baxter numbers enumerate in addition plane permutations (avoiding $2-14-3$). The problem of counting these permutations was open and has given rise to several conjectures, which we also prove in this paper. For each family (that of semi-Baxter -- or equivalently, plane -- and that of strong-Baxter permutations), we describe a generating tree, which translates into a functional equation for the generating function. For semi-Baxter permutations, it is solved using (a variant of) the kernel method: this gives an expression for the generating function while also proving its D-finiteness. From the obtained generating function, we derive closed formulas for the semi-Baxter numbers, a recurrence that they satisfy, as well as their asymptotic behavior. For strong-Baxter permutations, we show that their generating function is (a slight modification of) that of a family of walks in the quarter plane, which is known to be non D-finite.