ABSTRACT
Let denote the set consisting of integer sequences
such that
for all i, which are referred to as inversion sequences. In this paper, we enumerate members of
whose consecutive differences are bounded in three different ways:
,
and
for all i. In the first two cases, the corresponding subsets of
have cardinality given by the enumerator of the so-called Motzkin left-factors of length n−1 and by the Catalan number
for all
, respectively. In the third case, the subset of
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
.
Disclosure statement
No potential conflict of interest was reported by the author(s).