arXiv subjects

Benjamin Testart

Publications and source records attributed to Benjamin Testart.

5 recordsLinked to original sources

arXiv2026

Iterating the Lehmer code on inversion sequences: Catalan fixed points and finite stabilization

We study an operator $\Theta$ on finite integer sequences, where $\Theta(\sigma)_i$ counts the entries to the left of $\sigma_i$ that are strictly smaller than $\sigma_i$. This operator is a variant of the so-called Lehmer code. For every sequence $\sigma$, the image $\Theta(\sigma)$ is an inversion sequence, and the restriction of $\Theta$ to permutations of $[0,n-1]$ is a bijection onto inversion sequences of length $n$. We characterize the fixed points of $\Theta$ by avoidance of the pattern $101$ together with a saturation condition, prove that they are counted by the Catalan numbers, and give an explicit recursive bijection with Dyck paths. We also show that the sequences whose first $\Theta$-image is fixed are precisely those avoiding both $101$ and $201$. Finally, we prove finite stabilization for all inversion sequences, exhibit a family attaining the maximal stabilization time, and show that the second stabilization level is not closed under classical patterns.

math.CO↗

arXiv2026

On minimal pattern-containing inversion sequences

We introduce the notion of minimal inversion sequences for a pattern $\rho$, which form the smallest set of inversion sequences whose avoidance is equivalent to the avoidance of $\rho$ for inversion sequences. We give a characterization of $\rho$-minimal inversion sequences based on the occurrences of the pattern $\rho$ they contain, and use it to find upper and lower bounds on the lengths of $\rho$-minimal inversion sequences. We provide some enumerative results on the exact number of minimal inversion sequences for some patterns, through a bijection with increasing trees, and some exhaustive generation. Lastly, we enumerate inversion sequences which are equal to their reduction, and find an interesting connection with poly-Bernoulli numbers.

math.CO↗

arXiv2024

Generating trees growing on the left for pattern-avoiding inversion sequences

This work concerns a construction of pattern-avoiding inversion sequences from right to left we call the generating tree growing on the left. We first apply this construction to inversion sequences avoiding 201 and 210, resulting in a new way of computing their generating function. We then use a slightly modified construction to compute the generating function of inversion sequences avoiding 010 and 102, which was only conjectured before. These generating functions are algebraic in both instances. We end by discussing how the generating tree growing on the left can be applied in a more general setting.

math.CO↗

arXiv2024

Completing the enumeration of inversion sequences avoiding one or two patterns of length 3

We present four constructions of inversion sequences, and use them to compute the enumeration sequences of 24 classes of pattern-avoiding inversion sequences. This completes the enumeration of inversion sequences avoiding one or two patterns of length 3. Some of our constructions are based on generating trees. Others involve pattern-avoiding words, which we also count using generating trees. To solve some of these cases, we introduce a generalization of inversion sequences, which we call shifted inversion sequences. Lastly, we briefly discuss the asymptotics of pattern-avoiding inversion sequences, focusing on their exponential or super-exponential behavior.

math.CO↗

arXiv2022

Inversion sequences avoiding the pattern 010

Inversion sequences are integer sequences $(\sigma_1, \dots, \sigma_n)$ such that $0 \leqslant \sigma_i < i$ for all $1 \leqslant i \leqslant n$. The study of pattern-avoiding inversion sequences began in two independent articles by Mansour-Shattuck and Corteel-Martinez-Savage-Weselcouch in 2015 and 2016. These two initial articles solved the enumeration of inversion sequences avoiding a single pattern for every pattern of length 3 except the patterns 010 and 100. The case 100 was recently solved by Mansour and Yildirim. We solve the final case by making use of a decomposition of inversion sequences avoiding the pattern 010. Our decomposition needs to take into account the maximal value, and the number of distinct values occurring in the inversion sequence. We then expand our method to solve the enumeration of inversion sequences avoiding the pairs of patterns $\{010, 000\}, \{010, 110\}, \{010, 120\}$, and the Wilf-equivalent pairs $\{010, 201\} \sim \{010, 210\}$. For each family of pattern-avoiding inversion sequences considered, its enumeration requires the enumeration of some family of constrained words avoiding the same patterns, a question which we also solve.

math.CO↗