Browsing by Subject "Catalan numbers"
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Item Open Access An algorithmic approach based on generating trees for enumerating pattern-avoiding inversion sequences(Academic Press Ltd- Elsevier Science Ltd, 2023-05-18) Kotsireas, I.; Mansour, T.; Yıldırım, GökhanWe introduce an algorithmic approach based on a generating tree method for enumerating the inversion sequences with various pattern-avoidance restrictions. For a given set of patterns, we propose an algorithm that outputs either an accurate description of the succession rules of the corresponding generating tree or an ansatz. By using this approach, we determine the generating trees for the pattern classes In(000, 021), In(100, 021), In(110, 021), In(102, 021), In(100, 012), In(011, 201), In(011, 210) and In(120, 210). Then we use the kernel method, obtain generating functions of each class, and find enumerating formulas. Lin and Yan studied the classification of the Wilf-equivalences for inversion sequences avoiding pairs of length-three patterns and showed that there are 48 Wilf classes among 78 pairs. In this paper, we solve six open cases for such pattern classes. Moreover, we extend the algorithm to restricted growth sequences and apply it to several classes. In particular, we present explicit formulas for the generating functions of the restricted growth sequences that avoid either {12313, 12323}, {12313, 12323, 12333}, or {123 ··· 1}.Item Open Access Inversion sequences avoiding 021 and another pattern of length four(D M T C S, 2023-11-17) Toufik, M.; Yıldırım, GökhanWe study the enumeration of inversion sequences that avoid pattern 021 and another pattern of length four. We determine the generating trees for all possible pattern pairs and compute the corresponding generating functions. We introduce the concept of d-regular generating trees and conjecture that for any 021-avoiding pattern τ , the generating tree T ({021, τ }) is d-regular for some integer d.Item Open Access Pattern-avoiding permutations: The case of length three, four, and five(2023-06) Akbaş, ZilanA shorter permutation of length k is said to appear as a pattern in a longer per-mutation of length n if the longer permutation has a subsequence of length k that is order isomorphic to the shorter one. Otherwise, the longer permutation avoids the shorter one as a pattern. We use Sn(τ) to denote the set of permutations of length n that avoid pattern τ. Pattern avoidance induces an equivalence relation on the pattern set Sk. For ρ, τ ∈ Sk, we define the equivalence relation as follows: ρ ∼W τ if and only if |Sn(ρ)| = |Sn(τ)| for all n ≥ 1. The equivalence classes of this relation are called Wilf classes. The main questions are determining the Wilf classes of Sk and enumerating each class. We first study the Wilf classification and enumeration of each class for S3 and S4. We then present some new numerical results regarding the Wilf classification of pairs of patterns of length five. We define a Wilf class as small if it contains only one pair and big if it contains more than one pair. We show that there are at least 968 small Wilf classes and at most 13 big Wilf classes.