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### A cycle lemma for permutation inversions

S. Brunetti, A. Del Lungo, F. Del Ristoro
2002 Discrete Mathematics
Using 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  and for helpful comments.  ...

### Hurdles Hardly Have to Be Heeded [chapter]

Krister M. Swenson, Yu Lin, Vaibhav Rajan, Bernard M. E. Moret
2008 Lecture Notes in Computer Science
the probability of generating one in the process of sorting 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

Gracinda M. S. Gomes, John M. Howie
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].  ...

### On simultaneous conjugation of permutations [article]

Junyao Pan
2020 arXiv   pre-print
symmetric group, then there exists 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

Retno MAHARESI
2016 Journal of Applied Computer Science & Mathematics
Here, 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).  ...

### An O(n2) algorithm for maximum cycle mean of Monge matrices in max-algebra

Martin Gavalec, Ján Plávka
2003 Discrete Applied Mathematics
An O(n 2 ) algorithm is described 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.  ...

### Hurdles and Sorting by Inversions: Combinatorial, Statistical, and Experimental Results

Krister M. Swenson, Yu Lin, Vaibhav Rajan, Bernard M.E. Moret
2009 Journal of Computational Biology
In this paper, we study the probability of generating 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]

Eric Tannier, Marie-France Sagot
2004 Lecture Notes in Computer Science
We propose 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

Gary L. Ebert
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

Gary L. Ebert
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.  ...

### Spherically Symmetric Random Permutations [article]

2016 arXiv   pre-print
The extreme infinitely spherically symmetric 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]

Peter M. Higgins
2019 arXiv   pre-print
We investigate the question as to when the members of 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.  ...

### Visualizing the word problem, with an application to sixth groups

Carl Weinbaum
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

Peter M. Higgins
2013 Semigroup Forum
We investigate the question as to when the members of 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

Flora Dinkines
1951 Proceedings of the American Mathematical Society
Then xzx is of order 6 while yz;y = z.Therefore, 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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