Repetitions in graphs and sequences

Axenovich, Maria

doi:10.1007/978-3-319-24298-9_3

Cited by 5 publications

(6 citation statements)

References 34 publications

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“…Here is an example of a permutation of length 18 avoiding tight twins of length 3 or more (but we do not know how to generalize it for larger n): (14,15,16,3,2,1,10,11,12,5,4,18,8,9,17,7,6,13).…”

Section: Discussionmentioning

confidence: 99%

“…If a pair of twins jointly occupies a segment in π, then we call them tight twins. For example, in permutation (6, 5 , 7 , 1 , 2 , 3 , 4 , 9, 8), the red (5, 2, 3) and blue (7,1,4) subsequences form a pair of tight twins similar to permutation (3, 1, 2). Let tt(n) be the largest size of tight twins one can find in every n-permutation.…”

Section: Twins In Permutationsmentioning

confidence: 99%

“…To put our work in a broader context, we mention two similar problems: for graphs and for sequences. More can be found in a survey by Axenovich [4].…”

Section: Introductionmentioning

confidence: 99%

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Variations on Twins in Permutations

Dudek

Grytczuk²,

Ruciński³

2021

Electron. J. Combin.

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Let $\pi$ be a permutation of the set $[n]=\{1,2,\dots, n\}$. Two disjoint order-isomorphic subsequences of $\pi$ are called twins. How long twins are contained in every permutation? The well known Erdős-Szekeres theorem implies that there is always a pair of twins of length $\Omega(\sqrt{n})$. On the other hand, by a simple probabilistic argument Gawron proved that for every $n\geqslant 1$ there exist permutations with all twins having length $O(n^{2/3})$. He conjectured that the latter bound is the correct size of the longest twins guaranteed in every permutation. We support this conjecture by showing that almost all permutations contain twins of length $\Omega(n^{2/3}/\log n^{1/3})$. Recently, Bukh and Rudenko have tweaked our proof and removed the log-factor. For completeness, we also present our version of their proof (see Remark 2 below on the interrelation between the two proofs). In addition, we study several variants of the problem with diverse restrictions imposed on the twins. For instance, if we restrict attention to twins avoiding a fixed permutation $\tau$, then the corresponding extremal function equals $\Theta(\sqrt{n})$, provided that $\tau$ is not monotone. In case of block twins (each twin occupies a segment) we prove that it is $(1+o(1))\frac{\log n}{\log\log n}$, while for random permutations it is twice as large. For twins that jointly occupy a segment (tight twins), we prove that for every $n$ there are permutations avoiding them on all segments of length greater than $24$.

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Section: Discussionmentioning

confidence: 99%

Section: Twins In Permutationsmentioning

confidence: 99%

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Variations on Twins in Permutations

Dudek

Grytczuk²,

Ruciński³

2021

Electron. J. Combin.

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“…The answer in this case is 3n/4 + o k (n), and in particular the leading term is independent of k. Surprisingly, it turns out that this bound holds (with o k (n) replaced by O(1)) even if we wish to simultaneously decompose every graph on n vertices with the same number of edges [7]. We refer the reader to [16] for an interesting history of this problem, and mention that there are numerous variants which are actively studied (see, for instance, [2,5,10]).…”

Section: Introductionmentioning

confidence: 99%

Decomposing random permutations into order-isomorphic subpermutations

Groenland¹,

Johnston²,

Korándi³

et al. 2022

Preprint

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Two permutations σ and π are ℓ-similar if they can be decomposed into subpermutations σ (1) , . . . , σ (ℓ) and π (1) , . . . , π (ℓ) such that σ (i) is orderisomorphic to π (i) for all i ∈ [ℓ]. Recently, Dudek, Grytczuk and Ruciński posed the problem of determining the minimum ℓ for which two permutations chosen independently and uniformly at random are ℓ-similar. We show that two such permutations are O n 1/3 log 11/6 (n) -similar with high probability, which is tight up to a polylogarithmic factor. Our result also generalises to simultaneous decompositions of multiple permutations.

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“…In this paper we study this question for random words. To put our work in a broader perspective, we briefly discuss a couple of similar problems and results for graphs and permutations (see [3] for further examples).…”

Section: §1 Introductionmentioning

confidence: 99%

Long twins in random words

Dudek¹,

Grytczuk²,

Ruciński³

2021

Preprint

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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, with a computer-assisted proof, 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, for alphabets of size k ě 3, we get analogous lower bounds of the form 1.64 k`1 n, which are better than the known deterministic bounds for k ď 354. In addition, we present similar results for multiple twins in random words.

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Repetitions in graphs and sequences

Cited by 5 publications

References 34 publications

Variations on Twins in Permutations

Variations on Twins in Permutations

Decomposing random permutations into order-isomorphic subpermutations

Long twins in random words

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