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A cycle lemma for permutation inversions
2002
Discrete Mathematics
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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 [11] and for helpful comments. ...
doi:10.1016/s0012-365x(02)00254-6
fatcat:4q5y7zenwjepxmen3h23rs343i
Hurdles Hardly Have to Be Heeded
[chapter]
2008
Lecture Notes in Computer Science
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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. ...
doi:10.1007/978-3-540-87989-3_18
fatcat:4gydthscxjd2ta7apsqxrbpvwu
Nilpotents in finite symmetric inverse semigroups
1987
Proceedings of the Edinburgh Mathematical Society
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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]. ...
doi:10.1017/s0013091500026778
fatcat:icx3x6wmcnfwrl7g5duo6h6ure
On simultaneous conjugation of permutations
[article]
2020
arXiv
pre-print
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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 . ...
arXiv:1810.08971v3
fatcat:54edunvcwngjhhcir2wam4xv5i
Enumerating Hamiltonian Cycles in a Planar Graph Using Combinatorial Cycle Bases
2016
Journal of Applied Computer Science & Mathematics
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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). ...
doi:10.4316/jacsm.201601006
fatcat:63oljrl6xbfpvp4wn2mm5gdm7a
An O(n2) algorithm for maximum cycle mean of Monge matrices in max-algebra
2003
Discrete Applied Mathematics
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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. ...
doi:10.1016/s0166-218x(02)00395-5
fatcat:f5qtitao6bdfro727vwr4vmjbe
Hurdles and Sorting by Inversions: Combinatorial, Statistical, and Experimental Results
2009
Journal of Computational Biology
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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 ...
doi:10.1089/cmb.2009.0156
pmid:19747037
fatcat:owfgmvfueza2bn76vspvtfwwpm
Sorting by Reversals in Subquadratic Time
[chapter]
2004
Lecture Notes in Computer Science
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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. ...
doi:10.1007/978-3-540-27801-6_1
fatcat:xij3x3rnnzg6fjarras6vwze2q
Disjoint Circles: A Classification
1977
Transactions of the American Mathematical Society
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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. ...
doi:10.2307/1998927
fatcat:7dj2ba3nfnbwbj4nr3s3yjocbq
Disjoint circles: a classification
1977
Transactions of the American Mathematical Society
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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. ...
doi:10.1090/s0002-9947-1977-0442819-9
fatcat:l6nfuxjoijegdkl62yxz6id6sa
Spherically Symmetric Random Permutations
[article]
2016
arXiv
pre-print
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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. ...
arXiv:1611.01860v1
fatcat:ezsm6j2e65eq5ahrfwilff6xxu
Permutations of a semigroup that map to inverses
[article]
2019
arXiv
pre-print
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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. ...
arXiv:1901.05198v1
fatcat:ic2w2lxjcnafhlwpoi4xk2jqni
Visualizing the word problem, with an application to sixth groups
1966
Pacific Journal of Mathematics
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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. ...
doi:10.2140/pjm.1966.16.557
fatcat:5wcezsqorzhbrfoil43lwpiq4y
Permutations of a semigroup that map to inverses
2013
Semigroup Forum
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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. ...
doi:10.1007/s00233-013-9535-2
fatcat:inrorjw7wjcxbpart4yixzvhtq
Semi-automorphisms of symmetric and alternating groups
1951
Proceedings of the American Mathematical Society
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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. ...
doi:10.1090/s0002-9939-1951-0041140-8
fatcat:a4n2elxa4reargotpp5kqjzhq4
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