English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

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

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