Refined Restricted Inversion Sequences

Recently, the study of patterns in inversion sequences was initiated by Corteel–Martinez–Savage–Weselcouch and Mansour–Shattuck independently. Motivated their works a double Eulerian equidistribution due to Foata (1977), we investigate several classical statistics on restricted that are either known or conjectured be enumerated Catalan, Large Schröder, Baxter Euler numbers. One two highlights our results is fascinating bijection between 000-avoiding Simsun permutations, which together with Foata’s V- S-codes, provide proof equidistribution. The other one refinement conjecture Martinez Savage cardinality $${\mathbf{I}}_n(\ge ,\ge ,>)$$ n-th number, proved via so-called obstinate kernel method developed Bousquet-Mélou.