Order-Isomorphic Twins in Permutations

Bukh, Boris; Rudenko, Oleksandr

doi:10.1137/20m1323357

Cited by 12 publications

(21 citation statements)

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“…The proof of Gawron's upper bound comes from applying the first moment method to a uniformly random permutation, and there has since been work on finding a matching lower bound for random permutations. A lower bound of Ω n 2/3 / log 1/3 (n) was shown by Dudek, Grytczuk and Ruciński [13], and this was improved to the sharp bound of Ω(n 2/3 ) by Bukh and Rudenko [4]. There has also been further work of Dudek, Grytczuk and Ruciński in the same vein concerning k-twins [11], which are a collection of k pairwise disjoint orderisomorphic subpermutations of a single permutation, and other notions of twins with additional restrictions imposed or a weaker similarity condition (see [9,12]).…”

Section: A Simple O(mentioning

confidence: 87%

“…As observed in [15], the Erdős-Szekeres theorem provides a lower bound of Ω(n 1/2 ). This lower bound was improved to Ω(n 3/5 ) by Bukh and Rudenko [4].…”

Section: A Simple O(mentioning

confidence: 98%

“…In the following, the idea of constructing an auxiliary bipartite multigraph has previously been used by Dudek, Grytczuk and Ruciński [13] and later by Bukh and Rudenko [4] to study twins in a single random permutation. The use of a Poisson process to sample random permutations also appears in [4].…”

Section: Proof Of the Main Theoremsmentioning

confidence: 99%

“…With the n points ordered by their x-coordinate, the permutation can be read off as the relative ordering of the y-coordinates. This representation was used by Bukh and Rudenko [4]. Given sets of n "red" points and n "blue" points which define a pair of random permutations, we choose a matching between the red points and the blue points and only consider decompositions into subpermutations where the matched points are corresponding entries in the decomposition.…”

Section: Proof Of the Main Theoremsmentioning

confidence: 99%

“…The other main problem pertaining to (general) twins in permutations is to determine the maximum size of a twin that one is guaranteed to find in a permutation of length n. The best lower bound, due to Bukh and Rudenko [4], is currently Ω(n 3/5 ), while the best upper bound is O(n 2/3 ) obtained by applying a first moment calculation to a random permutation [15]. It would be interesting to narrow this gap.…”

Section: Open Problemsmentioning

confidence: 99%

See 4 more Smart Citations

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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Section: A Simple O(mentioning

confidence: 87%

“…As observed in [15], the Erdős-Szekeres theorem provides a lower bound of Ω(n 1/2 ). This lower bound was improved to Ω(n 3/5 ) by Bukh and Rudenko [4].…”

Section: A Simple O(mentioning

confidence: 98%

Section: Proof Of the Main Theoremsmentioning

confidence: 99%

Section: Proof Of the Main Theoremsmentioning

confidence: 99%

Section: Open Problemsmentioning

confidence: 99%

See 3 more Smart Citations

Decomposing random permutations into order-isomorphic subpermutations

Groenland¹,

Johnston²,

Korándi³

et al. 2022

Preprint

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Patterns in Ordered (random) Matchings

Dudek¹,

Grytczuk²,

Ruciński³

2022

LATIN 2022: Theoretical Informatics

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An ordered r-matching of size n is an r-uniform hypergraph on a linearly ordered set of vertices, consisting of n pairwise disjoint edges. Two ordered r-matchings are isomorphic if there is an order-preserving isomorphism between them. A pair of twins in an ordered r-matching is formed by two vertex disjoint isomorphic sub-matchings. Let t prq pnq denote the maximum size of twins one may find in every ordered r-matching of size n.By relating the problem to that of largest twins in permutations and applying some recent Erdős-Szekeres-type results for ordered matchings, we show that t prq pnq " Ω ´n 3 5¨p2 r´1 ´1q for every fixed r ě 2. On the other hand, t prq pnq " O ´n 2 r`1 ¯, by a simple probabilistic argument. As our main result, we prove that, for almost all ordered r-matchings of size n, the size of the largest twins achieves this bound.

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