Hereditary Semiorders and Enumeration of Semiorders by Dimension

In 2010, Bousquet-Mélou et al. defined sequences of nonnegative integers called ascent sequences and showed that the ascent sequences of length $n$ are in one-to-one correspondence with the interval orders, i.e., the posets not containing the poset $\mathbf{2}+\mathbf{2}$. Through the use of generating functions, this provided an answer to the longstanding open question of enumerating the (unlabeled) interval orders. A semiorder is an interval order having a representation in which all intervals have the same length. In terms of forbidden subposets, the semiorders exclude $\mathbf{2}+\mathbf{2}$ and $\mathbf{1}+\mathbf{3}$. The number of unlabeled semiorders on $n$ points has long been known to be the $n$th Catalan number. However, describing the ascent sequences that correspond to the semiorders under the bijection of Bousquet-Mélou et al. has proved difficult. In this paper, we discuss a major part of the difficulty in this area: the ascent sequence corresponding to a semiorder may have an initial subsequence that corresponds to an interval order that is not a semiorder. We define the hereditary semiorders to be those corresponding to an ascent sequence for which every initial subsequence also corresponds to a semiorder. We provide a structural result that characterizes the hereditary semiorders and use this characterization to determine the ordinary generating function for hereditary semiorders. We also use our characterization of hereditary semiorders and the characterization of semiorders of dimension $3$ given by Rabinovitch to provide a structural description of the semiorders of dimension at most $2$. From this description, we are able to determine the ordinary generating function for the semiorders of dimension at most $2$.

Ascent Sequences Avoiding Pairs of Patterns

Ascent sequences were introduced by Bousquet-Melou et al. in connection with (2+2)-avoiding posets and their pattern avoidance properties were first considered by Duncan and Steingrímsson. In this paper, we consider ascent sequences of length $n$ avoiding two patterns of length 3, and we determine an exact enumeration for 16 different pairs of patterns. Methods include simple recurrences, bijections to other combinatorial objects (including Dyck paths and pattern-avoiding permutations), and generating trees. We also provide an analogue of the Erdős-Szekeres Theorem to prove that any sufficiently long ascent sequence contains either many copies of the same number or a long increasing subsequence, with a precise bound.

Ascent sequences and Fibonacci numbers

An ascent sequence is one consisting of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it in the sequence. Ascent sequences have recently been shown to be related to (2+2)-free posets and a variety of other combinatorial structures. Let Fn denote the Fibonacci sequence given by the recurrence Fn = Fn-1 + Fn-2 if n ? 2, with F0 = 0 and F1 = 1. In this paper, we draw connections between ascent sequences and the Fibonacci numbers by showing that several pattern-avoidance classes of ascent sequences are enumerated by either Fn+1 or F2n-1. We make use of both algebraic and combinatorial methods to establish our results. In one of the apparently more difficult cases, we make use of the kernel method to solve a functional equation and thus determine the distribution of some statistics on the avoidance class in question. In two other cases, we adapt the scanning-elements algorithm, a technique which has been used in the enumeration of certain classes of pattern-avoiding permutations, to the comparable problem concerning pattern-avoiding ascent sequences.