Andrzej Dudek, Jarosław Grytczuk, Andrzej Ruciński
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引用次数: 1
Abstract
Twins in a finite word are formed by a pair of identical subwords placed at disjoint sets of positions. We investigate the maximum length of twins in a random word over a k-letter alphabet. The obtained lower bounds for small values of k significantly improve the best estimates known in the deterministic case. Bukh and Zhou in 2016 showed that every ternary word of length n contains twins of length at least 0.34n. Our main result states that in a random ternary word of length n, with high probability, one can find twins of length at least 0.41n. In the general case of alphabets of size \(k\geqslant 3\) we obtain analogous lower bounds of the form \(\frac{1.64}{k+1}n\) which are better than the known deterministic bounds for \(k\leqslant 354\). In addition, we present similar results for multiple twins in random words.
期刊介绍：
Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board.
The scope of Annals of Combinatorics is covered by the following three tracks:
Algebraic Combinatorics:
Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices
Analytic and Algorithmic Combinatorics:
Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms
Graphs and Matroids:
Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches
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