On minimal pattern-containing inversion sequences

We introduce the notion of minimal inversion sequences for a pattern $ρ$, which form the smallest set of inversion sequences whose avoidance is equivalent to the avoidance of $ρ$ for inversion sequences. We give a characterization of $ρ$-minimal inversion sequences based on the occurrences of the pattern $ρ$ they contain, and use it to find upper and lower bounds on the lengths of $ρ$-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.