Pattern avoidance in extensions of comb-like posets

Abstract

This paper investigates pattern avoidance in linear extensions of particular partially ordered sets (posets). Since the problem of enumerating pattern-avoiding linear extensions of posets without any additional restrictions is a very hard one, we focus on the class of posets called combs. A comb consists of a fully ordered spine and several fully ordered teeth, where the first element of each tooth coincides with a corresponding element of the spine. We consider two natural assignments of integers to elements of the combs; we refer to the resulting integer posets as type-α and type-β combs. In this paper, we enumerate the linear extensions of type-α and type-β combs that avoid some of the length-3 patterns w ∈ S 3 . Most notably, we shown the number of linear extensions of type-β combs that avoid 312 to be the same as the number 1 st+1 s(t+1) s of (t + 1)-ary trees on s nodes, where t is the length of each tooth and s is the length of the comb spine or, equivalently, the number of its teeth. We also investigate the enumeration of linear extensions of type-α and type-β combs that avoid multiple length-3 patterns simultaneously.

Poset pattern-avoidance problems posed by Yakoubov

“…The next theorem concerns linear extensions of β-labeled combs. Theorem 2.2 (Yakoubov,[13]). We have…”

“…Miklós Bóna's book Combinatorics of Permutations provides an excellent reference for anyone wishing to learn more about the flourishing area of research that deals with permutation patterns [4]. Recently, Sophia Yakoubov posed a natural question that links the study of linear extensions of posets with that of permutation patterns [13]. Suppose we are given a finite poset P with |P | = n. In addition, suppose we bijectively label the elements of P with the elements of [n].…”

“…E-mail address: cdefant@ufl.edu. Recently, Sophia Yakoubov posed a natural question that links the study of linear extensions of posets with that of permutation patterns [13]. Suppose we are given a finite poset P with |P | = n. In addition, suppose we bijectively label the elements of P with the elements of [n].…”

Pattern avoiding linear extensions of rectangular posets

“…The first and most fundamental open problem related to this work is to enumerate N E s,t (123) when s ≥ 3 and t ≥ 3. See Table 1 for the values of |N E s,t (123)| for some small s and t. Yakoubov [Yak15] also has some open problems involving monotone forbidden patterns of length three, so it would also be interesting to connect N E s,t (123) with one or more of these problems. Similarly, Levin, Pudwell, Riehl, and Sandberg [LPRS16] have some open problems involving binary heaps avoiding monotone patterns, and it would be interesting to connect N E s,t (123) with one or more of these problems.…”

“…In [Yak15] Yakoubov introduced an extensive new family of permutation enumeration problems. To state the most general of these problems, suppose n is a positive integer, ⇑ is a partial order on [n], and σ 1 , .…”