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A cycle lemma for permutation inversions

2002
*
Discrete Mathematics
*

Using

doi:10.1016/s0012-365x(02)00254-6
fatcat:4q5y7zenwjepxmen3h23rs343i
*a*"*cycle**lemma*",*a*combinatorial proof of the results is given. Moreover, some applications to the Gaussian binomial coe cient are illustrated. ... Some enumerative results concerning the*permutations*of the multiset {x m 1 1 ; x m 2 2 } with respect to the*inversion*parameter are established and it is shown that these depend on gcd(m1; m2). ... Acknowledgements The authors wish to thank the anonymous referees*for*suggesting the reference to Narayana's paper [11] and*for*helpful comments. ...##
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Hurdles Hardly Have to Be Heeded
[chapter]

2008
*
Lecture Notes in Computer Science
*

the probability of generating one in the process of sorting

doi:10.1007/978-3-540-87989-3_18
fatcat:4gydthscxjd2ta7apsqxrbpvwu
*a**permutation*if one does not take special precautions (as in*a*randomized algorithm,*for*instance). ... In this paper, we extend the work of Caprara and of Bergeron by providing simple and exact characterizations of the probability of encountering*a*hurdle or*a*fortress in*a*random*permutation*, as well as ... However, not every sorting*inversion**for**a**permutation*without hurdles is necessarily an oriented*inversion*; on the other hand, it is necessarily*a**cycle*-splitting*inversion*. ...##
###
Nilpotents in finite symmetric inverse semigroups

1987
*
Proceedings of the Edinburgh Mathematical Society
*

*Inverse*semigroups (see [3, Chapter V]) are of interest not only as

*a*naturally occurring special case of semigroups but also

*for*their role in describing partial symmetries. ... ,n -1) consists of all subpermutations of height r, i.e. all

*a*in SP n

*for*which |doma| = |ima| = r. Notice that J o consists solely of 0, the unique subpermutation with empty domain, and that ... Nilpotents in

*a*finite symmetric

*inverse*semigroup

*For*notation and

*for*basic properties of symmetric

*inverse*semigroups, see [3, Section V.I]. ...

##
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On simultaneous conjugation of permutations
[article]

2020
*
arXiv
*
pre-print

symmetric group, then there exists

arXiv:1810.08971v3
fatcat:54edunvcwngjhhcir2wam4xv5i
*a**permutation*in this symmetric inverting both of them. ... In this paper we give an affirmative answer to*a*conjecture proposed by Danny Neftin, that is, if the commutator of two*permutations*has at least n-4 fixed points where two*permutations*are in degree n ... Since there is no*a*transitive*cycle*factors chain of α on β, it follows that α −1 i α β j = 1*for*all i and j. By*Lemma*2.6 (i), there exist local*inverse*pairs between α −1 and α β . ...##
###
Enumerating Hamiltonian Cycles in a Planar Graph Using Combinatorial Cycle Bases

2016
*
Journal of Applied Computer Science & Mathematics
*

Here,

doi:10.4316/jacsm.201601006
fatcat:63oljrl6xbfpvp4wn2mm5gdm7a
*a*Hamiltonian*cycle*is represented as*a**permutation**cycle*of length n whose*permutation*and its corresponding*inverse**permutation*are not distinguished. ...*A*construction*for**a*class of Hamiltonian*cycles*in*a*complete graph consisting of n nodes is constructed based on ECO method. ... 4, ... on*a*complete graph K n, In Figure 2,*for*n = 4, the pair of*cycle**permutation*and its corresponding*inverse*is (2341, 4123). ...##
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An O(n2) algorithm for maximum cycle mean of Monge matrices in max-algebra

2003
*
Discrete Applied Mathematics
*

An O(n 2 ) algorithm is described

doi:10.1016/s0166-218x(02)00395-5
fatcat:f5qtitao6bdfro727vwr4vmjbe
*for*computing the maximum*cycle*mean (eigenvalue)*for*n × n matrices,*A*= (aij) fulÿlling Monge property, aij +*a*kl 6*a*il +*a*kj*for*any i ¡ k, j ¡ l. ...*A*similar result is presented*for*matrices with*inverse*Monge property. The standard algorithm*for*the general case works in O(n 3 ) time. ? (M. Gavalec), jan.plavka@tuke.sk (J. ... The proof of Theorem 2.1 is based on two*lemmas*.*Lemma*2.2. Let C = (i 1 ; i 2 ; : : : ; i k ; i 1 ) be*a**cycle*of length k ¿ 3. ...##
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Hurdles and Sorting by Inversions: Combinatorial, Statistical, and Experimental Results

2009
*
Journal of Computational Biology
*

In this paper, we study the probability of generating

doi:10.1089/cmb.2009.0156
pmid:19747037
fatcat:owfgmvfueza2bn76vspvtfwwpm
*a*hurdle in the process of sorting*a**permutation*if one does not take special precautions to avoid them (as in*a*randomized algorithm,*for*instance ... Using similar methods we provide the first asymptotically tight analysis of the probability that*a*fortress exists in*a*random*permutation*. ... ACKNOWLEDGMENTS*A*preliminary version of this work (Swenson et al., 2008) appeared in the proceedings of the 6th RECOMB Workshop on Comparative Genomics (RECOMB-CG'08), in volume 5267 of Lecture Notes ...##
###
Sorting by Reversals in Subquadratic Time
[chapter]

2004
*
Lecture Notes in Computer Science
*

We propose

doi:10.1007/978-3-540-27801-6_1
fatcat:xij3x3rnnzg6fjarras6vwze2q
*a*method*for*sorting*a*signed*permutation*by reversals in time O(n √ n log n). ... The problem of sorting*a*signed*permutation*by reversals is inspired by genome rearrangements in computational molecular biology. ... Fig. 3 . 3 Removal of*a**cycle*in the*permutation*ofFigure 2, and contraction of the same*cycle*in the*inverse**permutation*. ...##
###
Disjoint Circles: A Classification

1977
*
Transactions of the American Mathematical Society
*

*For*q

*a*prime-power, let IP(?) denote the miquelian

*inversive*plane of order q. The classification of certain translation planes of order q2, ...

*Lemma*(3. 2) implies that, if H contains

*a*3-

*cycle*, then H also contains

*a*2-

*cycle*. Hence, H can never be equal to A4. This is true

*for*type (I), (II), or (III) quadruples. ... The next

*lemma*tells us that, if H contains

*a*3-

*cycle*or

*a*4-

*cycle*, then H also contains

*a*2-

*cycle*, and hence the equivalence class of our quadruple contains

*a*quadruple of type (I) or (II) above. ...

##
###
Disjoint circles: a classification

1977
*
Transactions of the American Mathematical Society
*

*For*q

*a*prime-power, let IP(?) denote the miquelian

*inversive*plane of order q. The classification of certain translation planes of order q2, ...

*Lemma*(3. 2) implies that, if H contains

*a*3-

*cycle*, then H also contains

*a*2-

*cycle*. Hence, H can never be equal to A4. This is true

*for*type (I), (II), or (III) quadruples. ... The next

*lemma*tells us that, if H contains

*a*3-

*cycle*or

*a*4-

*cycle*, then H also contains

*a*2-

*cycle*, and hence the equivalence class of our quadruple contains

*a*quadruple of type (I) or (II) above. ...

##
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Spherically Symmetric Random Permutations
[article]

2016
*
arXiv
*
pre-print

The extreme infinitely spherically symmetric

arXiv:1611.01860v1
fatcat:ezsm6j2e65eq5ahrfwilff6xxu
*permutation*-valued processes are identified*for*the Hamming, Kendall-tau and Caley metrics. ... We consider random*permutations*which are spherically symmetric with respect to*a*metric on the symmetric group S_n and are consistent as n varies. ... If σ ∈ S n−1 is*a**permutation*with k*cycles*, then it has n − 1 extensions with the same number of*cycles*and 1 extension with k + 1*cycles*. ...##
###
Permutations of a semigroup that map to inverses
[article]

2019
*
arXiv
*
pre-print

We investigate the question as to when the members of

arXiv:1901.05198v1
fatcat:ic2w2lxjcnafhlwpoi4xk2jqni
*a*finite regular semigroup may be*permuted*in such*a*way that each member is mapped to one of its*inverses*. In general this is not possible. ... However we reformulate the problem in terms of*a*related graph and, using an application of Hall's Marriage*Lemma*, we show in particular that the finite full transformation semigroup does enjoy this property ... ACKNOWLEDGEMENT: The author would like to thank an anonymous referee of this paper*for*some enlightening observations that led to its improvement. ...##
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Visualizing the word problem, with an application to sixth groups

1966
*
Pacific Journal of Mathematics
*

*A*product of

*a*free generator and its

*inverse*is

*a*null word. If VISUALIZING THE WORD PROBLEM 559 VISUALIZING THE WORD PROBLEM 565 ... This leads to

*a*definition of W-+W (delete X x , , X n ) and W-+W (insert X % , , X x )

*for*n^l. ... The

*lemma*holds

*for*t = 2 with U = X, V -Y. Let t ^ 3 and assume the

*lemma*is true

*for*smaller *. , R t ) and X = R iy Y= R ό

*for*i < j. ...

##
###
Permutations of a semigroup that map to inverses

2013
*
Semigroup Forum
*

We investigate the question as to when the members of

doi:10.1007/s00233-013-9535-2
fatcat:inrorjw7wjcxbpart4yixzvhtq
*a*nite regular semigroup may be*permuted*in such*a*way that each member is mapped to one of its*inverses*. In general this is not possible. ... However we reformulate the problem in terms of*a*related graph and, using an application of Hall's Marriage*Lemma*, we show in particular that the nite full transformation semigroup does enjoy this property ... ACKNOWLEDGEMENT: The author would like to thank an anonymous referee of this paper*for*some enlightening observations that led to its improvement. ...##
###
Semi-automorphisms of symmetric and alternating groups

1951
*
Proceedings of the American Mathematical Society
*

Then xzx is of order 6 while yz;y = z.Therefore,

doi:10.1090/s0002-9939-1951-0041140-8
fatcat:a4n2elxa4reargotpp5kqjzhq4
*for**a*given 3-*cycle*in x, y contains this 3-*cycle*of its*inverse*, and conversely. ...*A**permutation*is*a*1-1 mapping of*a*set into itself. Definition 4.*For**a*given positive integer n, the finite symmetric group Sn is the set of all*permutations*of n objects. Definition 5. ...
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