Browsing by Subject "Generating trees"
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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 Generating tree method and applications to pattern-avoiding inversion sequences(2024-05) Gezer, MelisAn inversion sequence of length n is an integer sequence e = e1 · · · en such that 0 ≤ ei < i for each 0 ≤ i ≤ n. We use In to denote the set of inversion sequences of length n. Let [k] := {0, 1, · · · , k − 1} denote the alphabet and τ be a word of length k over this alphabet. A pattern of length k is simply a word over the alphabet [k]. We say an inversion sequence e ∈ In contains the pattern τ of length k if it contains a sub-sequence of length k that is order isomorphic to τ; otherwise, e avoids the pattern τ . For a given pattern τ , we use In(τ ) to denote the set of all τ -avoiding inversion sequences of length n. Firstly, we review the enumeration of inversion sequences that avoid patterns of length three. We then study an enumeration method based on generating trees and the kernel method to enumerate pattern-avoiding inversion sequences for general patterns. Then, we provide sampling algorithms for pattern-avoiding inversion sequences and apply them to some specific patterns. Based on extensive simulations, we study some statistics such as the number of zeros, the number of distinct elements, the number of repeated elements, and the maximum elements. Finally, we present a bijection between In(0312) and In(0321) that preserves these statistics.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.