Generating permutations with a given major index

In [S. Effler, F. Ruskey, A CAT algorithm for listing permutations with a given number of inversions, {\it I.P.L.}, 86/2 (2003)] the authors give an algorithm, which appears to be CAT, for generating permutations with a given major index. In the present paper we give a new algorithm for generating a Gray code for subexcedant sequences. We show that this algorithm is CAT and derive it into a CAT generating algorithm for permutations with a given major index

Median of an odd number of permutations

International audienceGiven m permutations π1, π2 . . . πm of {1, 2, . . . , n} and a distance function d, the median problem is to find a permutation π ∗ that is the "closest" of the m given permutations. Here, we study the problem under the Kendall-τ distance that counts the number of pairwise disagreements between permutations. This problem is also known, in the context of rank aggregation, as the Kemeny Score Problem and has been proved to be NP-hard when m ≥ 4. This article is an extension of [4], where some nice combinatorial properties of the case m = 3 where stated without proof, to the general case m ≥ 3, m odd

The cyclic sieving phenomenon: a survey

The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a 2004 paper. Let X be a finite set, C be a finite cyclic group acting on X, and f(q) be a polynomial in q with nonnegative integer coefficients. Then the triple (X,C,f(q)) exhibits the cyclic sieving phenomenon if, for all g in C, we have # X^g = f(w) where # denotes cardinality, X^g is the fixed point set of g, and w is a root of unity chosen to have the same order as g. It might seem improbable that substituting a root of unity into a polynomial with integer coefficients would have an enumerative meaning. But many instances of the cyclic sieving phenomenon have now been found. Furthermore, the proofs that this phenomenon hold often involve interesting and sometimes deep results from representation theory. We will survey the current literature on cyclic sieving, providing the necessary background about representations, Coxeter groups, and other algebraic aspects as needed.Comment: 48 pages, 3 figures, the sedcond version contains numerous changes suggested by colleagues and the referee. To appear in the London Mathematical Society Lecture Note Series. The third version has a few smaller change

Renormalization and Computation II: Time Cut-off and the Halting Problem

This is the second installment to the project initiated in [Ma3]. In the first Part, I argued that both philosophy and technique of the perturbative renormalization in quantum field theory could be meaningfully transplanted to the theory of computation, and sketched several contexts supporting this view. In this second part, I address some of the issues raised in [Ma3] and provide their development in three contexts: a categorification of the algorithmic computations; time cut--off and Anytime Algorithms; and finally, a Hopf algebra renormalization of the Halting Problem.Comment: 28 page

New algorithm for listing all permutations

The most challenging task dealing with permutation is when the element is large. In this paper, a new algorithm for listing down all permutations for n elements is developed based on distinct starter sets. Once the starter sets are obtained,each starter set is then cycled to obtain the first half of distinct permutations. The complete list of permutations is achieved by reversing the order of the first half of permutation. The new algorithm has advantages over the other methods due to its simplicity and easy to use