We prove that every permutation of a Cartesian product of two finite sets can be written as a composition of three permutations, the first of which only modifies the left projection, the second only the right projection, and the third again only the left projection, and three alternations is indeed the optimal number. We show that for two countably infinite sets, the corresponding optimal number of alternations, called the alternation diameter, is four. The notion of alternation diameter can be defined in any category. In the category of finite-dimensional vector spaces, the diameter is also three. For the category of topological spaces, we exhibit a single self-homeomorphism of the plane which is not generated by finitely many alternations of homeomorphisms that only change one coordinate. The results on finite sets and vector spaces were previously known in the context of memoryless computation.
The case of finite sets, which started this paper, is inspired by @cite_1 , where it is shown that any permutation of @math , for three sets @math with @math , can be written as a composition of finitely many permutations where alternately only @math or @math is permuted. Our notion of alternation diameter in the category of finite sets is related to this definition, as it also means refers to alternately permuting a product on the left and right''. The difference is that there is no communication coordinate (making it harder), but we allow the permutation to depend on the value on the right when permuting the value on the left, and vice versa (making it easier).
@cite_1
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Distributions over permutations arise in applications ranging from multi-object tracking to ranking of instances. The difficulty of dealing with these distributions is caused by the size of their domain, which is factorial in the number of considered entities ( @math ). It makes the direct definition of a multinomial distribution over permutation space impractical for all but a very small @math . In this work we propose an embedding of all @math permutations for a given @math in a surface of a hypersphere defined in @math . As a result of the embedding, we acquire ability to define continuous distributions over a hypersphere with all the benefits of directional statistics. We provide polynomial time projections between the continuous hypersphere representation and the @math -element permutation space. The framework provides a way to use continuous directional probability densities and the methods developed thereof for establishing densities over permutations. As a demonstration of the benefits of the framework we derive an inference procedure for a state-space model over permutations. We demonstrate the approach with applications. where (an)n ENo is some sequence of nonnegative numbers, (Sn),nENo is the sequence of partial sums, S0 = 0, Sn = XflXk, of another sequence (Xk)kEN of i.i.d. random variables, and A c R is a fixed Borel set such as [0,1] or [0, oo). Examples of such convolution series are subordinated distributions (f=0Oan = 1) which arise as distributions of random sums, and harmonic and ordinary renewal measures (a0 = 0, an = 1 n for all n C N in the first, an = 1 for all n C NO in the second case). These examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes, they are also of interest in connection with representation theorems such as the Levy representation of infinitely divisible distributions. A traditional approach to such problems is via regular variation: If the underlying random variables are nonnegative we can use Laplace transforms and the related Abelian and Tauberian theorems [see, e.g., Stam (1973) in the context of subordination and Feller (1971, XIV.3) in connection with renewal theory; Embrechts, Maejima, and Omey (1984) is a recent treatment of generalized renewal measures along these lines]. The approach of the present paper is based on the Wiener-Levy-Gel'fand theorem and has occasionally been called the Banach algebra method. In Gruibel (1983) we gave a new variant of this method for the special case of lattice distributions, showing that by using the appropriate Banach algebras of sequences, arbitrarily fine expansions are possible under certain assumptions on the higher-order differences of (P(X1 = n))fnEN. Here we give a corresponding treatment of nonlattice distributions. We restrict ourselves to an analogue of first-order differences and obtain a number of theorems which perhaps are described best as next-term results. To explain this let us consider a special case in more detail. This paper considers the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices. For any given @math binary matrix @math , the GM-MDS conjecture, proposed by , states that if @math satisfies the so-called MDS condition, then for any field @math of size @math , there exists an @math MDS code whose generator matrix @math , with entries in @math , fits the matrix @math (i.e., @math is the support matrix of @math ). Despite all the attempts by the coding theory community, this conjecture remains still open in general. It was shown, independently by and , that the GM-MDS conjecture holds if the following conjecture, referred to as the TM-MDS conjecture, holds: if @math satisfies the MDS condition, then the determinant of a transform matrix @math , such that @math fits @math , is not identically zero, where @math is a Vandermonde matrix with distinct parameters. In this work, we first reformulate the TM-MDS conjecture in terms of the Wronskian determinant, and then present an algebraic-combinatorial approach based on polynomial-degree reduction for proving this conjecture. Our proof technique's strength is based primarily on reducing inherent combinatorics in the proof. We demonstrate the strength of our technique by proving the TM-MDS conjecture for the cases where the number of rows ( @math ) of @math is upper bounded by @math . For this class of special cases of @math where the only additional constraint is on @math , only cases with @math were previously proven theoretically, and the previously used proof techniques are not applicable to cases with @math . We study the length of the longest increasing and longest decreasing subsequences of random permutations drawn from the Mallows measure. Under this measure, the probability of a permutation ( S_n ) is proportional to (q^ Inv ( ) ) where (q ) is a real parameter and ( Inv ( ) ) is the number of inversions in ( ). The case (q=1 ) corresponds to uniformly random permutations. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. We determine the typical order of magnitude of the lengths of the longest increasing and decreasing subsequences, as well as large deviation bounds for them. We also provide a simple bound on the variance of these lengths, and prove a law of large numbers for the length of the longest increasing subsequence. Assuming without loss of generality that (q<1 ), our results apply when (q ) is a function of (n ) satisfying (n(1-q) ). The case that (n(1-q)=O(1) ) was considered previously by Mueller and Starr. In our parameter range, the typical length of the longest increasing subsequence is of order (n 1-q ), whereas the typical length of the longest decreasing subsequence has four possible behaviors according to the precise dependence of (n ) and (q ). We show also that in the graphical representation of a Mallows-distributed permutation, most points are found in a symmetric strip around the diagonal whose width is of order (1 (1-q) ). This suggests a connection between the longest increasing subsequence in the Mallows model and the model of last passage percolation in a strip.