Even Permutations Research Articles

Realization of even permutations of even degree by products of four involutions without fixed points

Abstract We consider representations of an arbitrary permutation π of degree 2n, n ⩾ 3, by products of the so-called (2 n )-permutations (any cycle of such a permutation has length 2). We show that any even permutation is represented by the product of four (2 n )-permutations. Products of three (2 n )-permutations cannot represent all even permutations. Any odd permutation is realized (for odd n) by a product of five (2 n )-permutations.

A decomposition of ballot permutations, pattern avoidance and Gessel walks

A permutation whose any prefix has no more descents than ascents is called a ballot permutation. In this paper, we present a decomposition of ballot permutations that enables us to construct a bijection between ballot permutations and odd order permutations, which proves a set-valued extension of a conjecture due to Spiro using the statistic of peak values. This bijection also preserves the neighbors of the largest letter in permutations and thus resolves a refinement of Spiro's conjecture proposed by Wang and Zhang. Our decomposition can be extended to well-labeled positive paths, a class of generalized ballot permutations arising from polytope theory, that were enumerated by Bernardi, Duplantier and Nadeau.We will also investigate the enumerative aspect of ballot permutations avoiding a single pattern of length 3 and establish a connection between 213-avoiding ballot permutations and Gessel walks.

Bijective Cremona transformations of the plane

We study the birational self-maps of the projective plane over finite fields that induce permutations on the set of rational points. As a main result, we prove that no odd permutation arises over a non-prime finite field of characteristic two, which completes the investigation initiated by Cantat about which permutations can be realized this way. Main ingredients in our proof include the invariance of parity under groupoid conjugations by birational maps, and a list of generators for the group of such maps.

The peak and descent statistics over ballot permutations

A ballot permutation is a permutation π such that in any prefix of π the descent number is not more than the ascent number. By using a reversal-concatenation map, we (i) give a formula for the joint distribution (pk, des) of the peak and descent statistics over ballot permutations, (ii) connect this distribution and the joint distribution (pk, des) over ordinary permutations in terms of generating functions, and (iii) confirm Spiro's conjecture which finds the equidistribution of the descent statistic for ballot permutations and an analogue of the descent statistic for odd order permutations.

Log-concavity of the excedance enumerators in positive elements of Type A and Type B Coxeter Groups

The classical Eulerian Numbers An,k are known to be log-concave. Let Pn,k and Qn,k be the number of even and odd permutations with k excedances. In this paper, we show that Pn,k and Qn,k are log-concave. For this, we introduce the notion of strong synchronisation and ratio-alternating which are motivated by the notion of synchronisation and ratio-dominance, introduced by Gross, Mansour, Tucker and Wang in 2014.We show similar results for Type B Coxeter Groups. We finish with some conjectures to emphasise the following: though strong synchronisation is stronger than log-concavity, many pairs of interesting combinatorial families of sequences seem to satisfy this property.

On the character tables of symmetric groups

In this paper, some zeros and non-zeros in the character tables of symmetric groups are displayed in the partition forms. In particular, more zeros of self conjugate partitions beside odd permutations are heavily investigated.

A Toeplitz Property of Ballot Permutations and Odd Order Permutations

We give a new semi-combinatorial proof for the equality of the number of ballot permutations of length $n$ and the number of odd order permutations of length $n$, which was originally proven by Bernardi, Duplantier and Nadeau. Spiro conjectures that the descent number of ballot permutations and certain cyclic weights of odd order permutations of the same length are equi-distributed. We present a bijection to establish a Toeplitz property for ballot permutations with any fixed number of descents, and a Toeplitz property for odd order permutations with any fixed cyclic weight. This allows us to refine Spiro's conjecture by tracking the neighbors of the largest letter in permutations.

Ballot permutations and odd order permutations

A permutation π is ballot if, for all k, the word π1⋯πk has at least as many ascents as it has descents. Let b(n) denote the number of ballot permutations of size n, and let p(n) denote the number of permutations which have odd order in the symmetric group Sn. Callan conjectured that b(n)=p(n) for all n, which was proved by Bernardi, Duplantier, and Nadeau.We propose a refinement of Callan’s original conjecture. Let b(n,d) denote the number of ballot permutations with d descents. Let p(n,d) denote the number of odd order permutations with M(π)=d, where M(π) is a certain statistic related to the cyclic descents of π. We conjecture that b(n,d)=p(n,d) for all n and d. We prove this stronger conjecture for the cases d=1,2,3, and d=⌊(n−1)∕2⌋, and in each of these cases we establish formulas for b(n,d) involving Eulerian numbers and Eulerian–Catalan numbers.

Some properties of the cycle decomposition of WG-NLFSR

<p style='text-indent:20px;'>In this paper, we give some properties of the cycle decomposition of a nonlinear feedback shift register called WG-NLFSR which was presented by Mandal and Gong recently. First we give the parity of the state transition transformation of WG-NLFSR and then by the relation of the parity of a permutation and its number of cycles given in Theorem 2 in Section 1, we show that the number of cycles in the cycle decomposition of WG-NLFSR is even. Second we study the properties of the cycle decomposition of WG-NLFSR when the coefficients of the characteristic polynomial belong to the proper subfields of the finite field on which the WG-NLFSR is defined. Finally, we give some properties of the cycle decomposition of the filtering WG7-NLFSR.

Odd extensions of transitive groups via symmetric graphs – The cubic case

When dealing with symmetry properties of mathematical objects, one of the fundamental questions is to determine their full automorphism group. In this paper this question is considered in the context of even/odd permutations dichotomy. More precisely: when is it that the existence of automorphisms acting as even permutations on the vertex set of a graph, called even automorphisms, forces the existence of automorphisms that act as odd permutations, called odd automorphisms. As a first step towards resolving the above question, complete information on the existence of odd automorphisms in cubic symmetric graphs is given.

The functional graph of linear maps over finite fields and applications

Let $$\mathbb F_{q}$$ be the finite field with q elements and $$n\ge 2$$ be a positive integer. We study the functional graph associated to linear maps over finite fields. In particular, we describe the functional graph $$\mathcal {G}_f(\mathbb F_{q^n})$$ associated to the map induced by $$L_f$$ on $$\mathbb F_{q^n}$$ , where f is any irreducible divisor of $$x^n-1$$ over $$\mathbb F_q$$ and $$L_f$$ is the q-associate of f. This description derives interesting information on the graph $$\mathcal {G}_f(\mathbb F_{q^n})$$ , such as the number of cycles and the average of the preperiod length. When $$\gcd (f, x^n-1)=1$$ , $$L_f$$ is a permutation on $$\mathbb F_{q^n}$$ and the cycle decomposition of $$\mathcal {G}_f(\mathbb F_{q^n})$$ is well known. In this case, we present some applications of this result, such as the construction of linear involutions over odd characteristic and permutations with few fixed points.

An algorithm for generating permutation algebras using soft spaces

ABSTRACTSoft set theory has recently gained significance for finding rational and logical solutions to various real-life problems, which involve uncertainty, impreciseness and vagueness. In this paper, we introduced an algorithm to find permutation algebras using soft topological space. Moreover, this class of permutation algebras is called even (odd) permutation algebras if its permutation is even (odd). Furthermore, new concepts in permutation algebras are investigated such as splittable permutation algebra and ambivalent permutation algebra. Furthermore, several examples are given to illustrate the concepts introduced in this paper.

Sign of Permutation Induced by Nagata Automorphism over Finite Fields

This paper proves that the Nagata automorphism over a finite field can be mimicked by a tame automorphism which is a composition of four elementary automorphisms. By investigating the sign of the permutations induced by the above elementary automorphisms, one can see that if the Nagata automorphism is defined over a prime field of characteristic two, the Nagata automorphism induces an odd permutation, and otherwise, the Nagata automorphism induces an even permutation.

Sign of Permutations Induced by Anick and Nagata-Anick Automorphisms over Finite Fields

In this paper, we investigate the sign of permutations induced by the Anick automorphism and the Nagata-Anick automorphism over finite fields. We shall prove that if the Anick automorphism and the Nagata-Anick automorphism are defined over a prime field of characteristic two, they induce odd permutations, and otherwise, they induce even permutations.

On the linear classification of even and odd permutation matrices and the complexity of computing the permanent

The problem of linear classification of the parity of permutation matrices is studied. This problem is related to the analysis of complexity of a class of algorithms designed for computing the permanent of a matrix that generalizes the Kasteleyn algorithm. Exponential lower bounds on the magnitude of the coefficients of the functional that classifies the even and odd permutation matrices in the case of the field of real numbers and similar linear lower bounds on the rank of the classifying map for the case of the field of characteristic 2 are obtained.

Area functions on affine planes

Oriented area functions are functions defined on the set of ordered triangles of an affine plane which are antisymmetric under odd permutations of the vertices and which behave additively when triangles are cut into two. We compare several elementary properties which such an area function may have (roughly speaking shear invariance, equality of area of the two triangles obtained by cutting a parallelogram along a diagonal, and equality of area of the two triangles obtained by cutting a triangle along a median). It turns out purely by arguments of elementary affine geometry (if cleverly arranged) that these properties are grosso modo equivalent, although one has to be careful about “pathological” situations. Furthermore, all oriented area functions satisfying these properties are explicitly determined. Finally they are compared with so-called geometric valuations.

Counting of even and odd restricted permutations

Let p be a permutation of the set N n = {1, 2, …, n } . We introduce techniques for counting N ( n ; k ; r ; I ; π ) , the number of even or odd restricted permutations of N n satisfying the conditions − k ≤ p ( i ) − i ≤ r (for arbitrary natural numbers k and r ) and p ( i ) − i ∉ I (for some set I ) and π = 0 for even permutations and π = 1 for odd permutations.

On products of long cycles: short cycle dependence and separation probabilities

We present various results on multiplying cycles in the symmetric group. One result is a generalisation of the following theorem of Boccara (Discret Math 29:105---134, 1980): the number of ways of writing an odd permutation in the symmetric group on $$n$$n symbols as a product of an $$n$$n-cycle and an $$(n-1)$$(n-1)-cycle is independent of the permutation chosen. We give a number of different approaches of our generalisation. One partial proof uses an inductive method which we also apply to other problems. In particular, we give a formula for the distribution of the number of cycles over all products of cycles of fixed lengths. Another application is related to the recent notion of separation probabilities for permutations introduced by Bernardi et al. (Comb Probab Comput 23:201---222, 2014).

Determining the parity of a permutation using an experimental NMR qutrit

We present the NMR implementation of a recently proposed quantum algorithm to find the parity of a permutation. In the usual qubit model of quantum computation, speedup requires the presence of entanglement and thus cannot be achieved by a single qubit. On the other hand, a qutrit is qualitatively more quantum than a qubit because of the existence of quantum contextuality and a single qutrit can be used for computing. We use the deuterium nucleus oriented in a liquid crystal as the experimental qutrit. This is the first experimental exploitation of a single qutrit to carry out a computational task.

Sign Under the Domino Robinson-Schensted Maps

We generalize a formula obtained independently by Reifegerste and Sjostrand for the sign of a permutation under the classical Robinson-Schensted map to a family of domino Robinson-Schensted algorithms.