# Representing Permutations with Few Moves

@article{Bereg2016RepresentingPW, title={Representing Permutations with Few Moves}, author={Sergey Bereg and Alexander E. Holroyd and Lev Nachmanson and Sergey Pupyrev}, journal={SIAM J. Discret. Math.}, year={2016}, volume={30}, pages={1950-1977} }

Consider a finite sequence of permutations of the elements 1,...,n, with the property that each element changes its position by at most 1 from any permutation to the next. We call such a sequence a tangle, and we define a move of element i to be a maximal subsequence of at least two consecutive permutations during which its positions form an arithmetic progression of common difference +1 or -1. We prove that for any initial and final permutations, there is a tangle connecting them in which each…

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## References

SHOWING 1-10 OF 26 REFERENCES

Sorting a bridge hand

- Computer Science, MathematicsDiscret. Math.
- 2001

A complete answer to the optimal sorting problem of a permutation of length n in at most 3n/4 moves is given, namely [(n + 1)/2].

Drawing Permutations with Few Corners

- Mathematics, Computer ScienceGraph Drawing
- 2013

This work addresses the problem of minimizing the number of crossings together with theNumber of corners of the paths, focusing on classes of permutations in which both can be minimized simultaneously.

Patterns in Permutations and Words

- Mathematics, Computer ScienceMonographs in Theoretical Computer Science. An EATCS Series
- 2011

The author collects the main results in the field in this up-to-date, comprehensive reference volume and highlights significant achievements in the area, and points to research directions and open problems.

Compositions of pattern restricted sets of permutations

- Mathematics, Computer ScienceAustralas. J Comb.
- 2007

The composition of two pattern restricted classes X,Y is the set of all permutation products θφ where θ ∈ X,φ ∈ Y . This set is also defined by pattern restrictions. Examples are given where this set…

Enumerative combinatorics

- Mathematics, Computer ScienceSIGA
- 2008

This review of 3 Enumerative Combinatorics, by Charalambos A.good, does not support this; the label ‘Example’ is given in a rather small font followed by a ‘PROOF,’ and the body of an example is nonitalic, utterly unlike other statements accompanied by demonstrations.

Enumerative Combinatorics: Volume 1

- Mathematics
- 2011

Richard Stanley's two-volume basic introduction to enumerative combinatorics has become the standard guide to the topic for students and experts alike. This thoroughly revised second edition of…

RC-Graphs and Schubert Polynomials

- Mathematics, Computer ScienceExp. Math.
- 1993

A new set of diagrams that encode the Schubert polynomials are introduced using a formula of Billey, Jockusch and Stanley, Fomin and Kirillov and a new proof of Monk’s rule is given using an insertion algorithm on rc-graphs.

Random sorting networks

- Mathematics
- 2007

Abstract A sorting network is a shortest path from 12 ⋯ n to n ⋯ 21 in the Cayley graph of S n generated by nearest-neighbour swaps. We prove that for a uniform random sorting network, as n → ∞ the…

On the number of arrangements of pseudolines

- Computer Science, MathematicsSCG '96
- 1996

The vector (τ1, τ2, ..., τ_n) is shown to be an encoding for the arrangement of n pseudolines in the Euclidean plane.

Sorting inc logn parallel steps

- Mathematics
- 1983

We give a sorting network withcn logn comparisons. The algorithm can be performed inc logn parallel steps as well, where in a parallel step we comparen/2 disjoint pairs. In thei-th step of the…