Enumeration of smooth inversion sequences and proof of a recent related conjecture

Let (Formula presented.) denote the set consisting of integer sequences (Formula presented.) such that (Formula presented.) for all i, which are referred to as inversion sequences. In this paper, we enumerate members of (Formula presented.) whose consecutive differences are bounded in three different ways: (Formula presented.), (Formula presented.) and (Formula presented.) for all i. In the first two cases, the corresponding subsets of (Formula presented.) have cardinality given by the enumerator of the so-called Motzkin left-factors of length n−1 and by the Catalan number (Formula presented.) for all (Formula presented.), respectively. In the third case, the subset of (Formula presented.) is equinumerous with the set of rooted tandem duplication trees on n gene segments, which arise in applications to DNA research. Using our results from this case, we establish a conjecture concerning the enumerator of a certain class of Catalan restricted growth sequences. Finally, new polynomial generalizations of the underlying counting sequences are obtained in the first two cases above by considering the joint distribution of parameters on the corresponding subsets of (Formula presented.).