A remark on linear permutation polynomials

We give an explicit representation of the class of linear permutation polynomials. By the representation, the number of them can be computed easily.     

Graceful signed graphs: II. The case of signed cycles with connected negative sections

In our earlier paper [9], generalizing the well known notion of graceful graphs, a (p, m, n)-signed graph S of order p, with m positive edges and n negative edges, is called graceful if there exists an injective function f that assigns to its p vertices integers 0, 1,...,q = m + n such that when to each edge uv of S one assigns the absolute difference |f(u)-f(v)| the set of integers received by the positive edges of S is {1,2,...,m} and the set of integers received by the negative edges of S is {1,2,...,n}. Considering the conjecture therein that all signed cycles Z_k, of admissible length k 3 and signed structures, are graceful, we establish in this paper its truth for all possible signed cycles of lengths 0, 2 or 3 (mod 4) in which the set of negative edges forms a connected subsigraph.     

Alternating signs of quiver coefficients

We prove a formula for the Grothendieck class of a quiver variety, which generalizes the cohomological component formulas of Knutson, Miller, and Shimozono. Our formula implies that the -theoretic quiver coefficients have alternating signs and gives an explicit combinatorial formula for these coefficients. We also prove some new variants of the factor sequences conjecture and a conjecture of Knutson, Miller, and Shimozono, which states that their double ratio formula agrees with the original quiver formulas of the author and Fulton. For completeness we include a short proof of the ratio formula.

     

On the Movement of Transitive Permutation p-Groups

Let G be a transitive permutation group on a set and m a positive integer. If | – | m for every subset of and all g G, then || 2mp/(p – 1) where p is the least odd prime dividing |G|. It was shown by Mann and Praeger [13] that, for p = 3, the 3-groups G which attain this bound have exponent p. In this paper we will show a generalization of this result for any odd primes.AMS Subject Classification (2000), 20BXX     

Periodic Groups Covered by Transitive Subgroups of Finitary Permutations or by Irreducible Subgroups of Finitary Transformations

Let be either the class of all transitive groups of finitary permutations, or the class of all periodic irreducible finitary linear groups. We show that almost primitive -groups are countably recognizable, while totally imprimitive -groups are in general not countably recognizable. In addition we derive a structure theorem for groups all of whose countable subsets are contained in totally imprimitive -subgroups. It turns out that totally imprimitive -groups in the class are countably recognizable.

     

Signed domination numbers of directed graphs

The concept of signed domination number of an undirected graph (introduced by J. E. Dunbar, S. T. Hedetniemi, M. A. Henning and P. J. Slater) is transferred to directed graphs. Exact values are found for particular types of tournaments. It is proved that for digraphs with a directed Hamiltonian cycle the signed domination number may be arbitrarily small.     

Transitive permutation groups and equipotent voting rules

Let F a two-alternative voting rule and G_F the subgroup of permutations of the voters under which F is invariant. Group theoretic properties of G_F provide information about the voting rule F. In particular, sets of imprimitivity of G_F describe the ‘committee decomposition’ structure of F and permutation group transitivity of G_F (equipotency) is shown to be closely connected with equal distribution of power among the voters. If equipotency replaces anonymity in the hypotheses of May's theorem, voting rules other than simple majority are possible. By combining equipotency with two additional social choice conditions a new characterization of simple majority rule is obtained. Equipotency is proposed as an important alternative to the more restrictive anonymity as a fairness criterion in social choice.     

The peak algebra and the descent algebras of types B and D

We show the existence of a unital subalgebra of the symmetric group algebra linearly spanned by sums of permutations with a common peak set, which we call the peak algebra. We show that is the image of the descent algebra of type B under the map to the descent algebra of type A which forgets the signs, and also the image of the descent algebra of type D. The algebra contains a two-sided ideal which is defined in terms of interior peaks. This object was introduced in previous work by Nyman (2003); we find that it is the image of certain ideals of the descent algebras of types B and D. We derive an exact sequence of the form . We obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs. In particular, we construct two new commutative semisimple subalgebras of the descent algebra (of dimensions and by grouping permutations according to their number of peaks or interior peaks. We discuss the Hopf algebraic structures that exist on the direct sums of the spaces and over and explain the connection with previous work of Stembridge (1997); we also obtain new properties of his descents-to-peaks map and construct a type B analog.

     

Groups operating on fuzzy sets

In crisp environment the notions of normal subgroup and group operating on a set are well known due to many applications. In this paper we study extensions of these classical notions to the larger universe of fuzzy sets. We obtain a characterization of operations of a fuzzy group on a fuzzy set in terms of homomorphisms of crisp groups.