An Upper Bound on the Number of Circular Transpositions to Sort a Permutation

Zuylen, Anke van; Bieron, James C.; Schalekamp, Frans; Yu, Gexin

doi:10.48550/arxiv.1402.4867

Cited by 2 publications

(3 citation statements)

References 7 publications

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“…In the case of sorting the permutation π(x) = x + ⌊N/2⌋, at least N 2 /4 swaps are required due to the distance of each number from its destination. This has recently been proved to be the worst case [vZBSY14]. By the above discussion, writhe considerations easily yield a very close bound of (N − 1) 2 /4 transpositions, for π(x) = N − x.…”

Section: Extreme Valuesmentioning

confidence: 62%

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The writhe of permutations and random framed knots

Even-Zohar

2016

Random Struct. Alg.

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We introduce and study the writhe of a permutation, a circular variant of the well-known inversion number. This simple permutation statistics has several interpretations, which lead to some interesting properties. For a permutation sampled uniformly at random, we study the asymptotics of the writhe, and obtain a non-Gaussian limit distribution.This work is motivated by the study of random knots. A model for random framed knots is described, which refines the Petaluma model, studied in [EZHLN16]. The distribution of the framing in this model is equivalent to the writhe of random permutations.

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Section: Extreme Valuesmentioning

confidence: 62%

“…In Section 3 we discuss the relation of the writhe to other combinatorial notions, such as the alternating inversion number [Che08], and circular bubble sort [Jer85,vZBSY14]. We also fit the writhe into the context of nonparametric statistics for circular rank correlation [FL82,Fis95].…”

Section: Introductionmentioning

confidence: 99%

The writhe of permutations and random framed knots

Even-Zohar

2016

Random Struct. Alg.

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“…Remark 4. 10. The upper bound of n 2 − p on diam(H α ) in Theorem 4.9 corresponds to the number of pairs {i, j} with i < j and i, j ∈ [n] that are incomparable with respect to the poset P = ([n], ≤ α ).…”

Section: Path Graphsmentioning

confidence: 99%

Diameters of Components of Friends-and-Strangers Graphs are Not Polynomially Bounded

Ryan¹

2022

Preprint

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Given two graphs X and Y on n vertices, the friends-and-strangers graph FS(X, Y ) has as its vertices all n! bijections from V (X) to V (Y ), with bijections σ, τ adjacent if and only if they differ on two elements of V (X), whose mappings are adjacent in Y . In this work, we study the diameters of friends-and-strangers graphs, which correspond to the largest number of swaps necessary to achieve one configuration from another. We provide families of constructions XL and YL for all integers L ≥ 1 to show that diameters of connected components of friends-and-strangers graphs fail to be polynomially bounded in the size of X and Y , resolving a question raised by Alon, Defant, and Kravitz in the negative. Specifically, our construction yields that there exist infinitely many values of n for which there are n-vertex graphs X and Y with the diameter of a component of FS(X, Y ) at least n (log n)/(log log n) . We also study the diameters of components of friends-and-strangers graphs when X is taken to be a path graph or a cycle graph, showing that any component of FS(Pathn, Y ) has diameter at most |E(Y )|, and diam(FS(Cycle n , Y )) is O(n 3 ) whenever FS(Cycle n , Y ) is connected. We conclude the work with several conjectures that aim to generalize this latter result.

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An Upper Bound on the Number of Circular Transpositions to Sort a Permutation

Cited by 2 publications

References 7 publications

The writhe of permutations and random framed knots

The writhe of permutations and random framed knots

Diameters of Components of Friends-and-Strangers Graphs are Not Polynomially Bounded

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