A new class of Gelfand pairs

The well-known result for rank 1 real symmetric spacesX=G/H, stating the multiplicity-free decomposition of the natural action ofG onL ²(X), is extended to the classical rank 1p-adic symmetric spaces. The absence of ap-adic analogue of the Laplace-Beltrami operator causes additional complications and necessitates a global study of the invariant distributions on the quotient space.     

A class of permutation quadrinomials

In this paper, we investigate the permutation behavior of a class of quadrinomials. Each term of these quadrinomials has a Niho-type exponent, and two sets of coefficient triples making the quadrinomials to be permutations are obtained. We use a substitution to transform the permutation problem into the root distribution problem in the unit circle of certain quadratic and cubic equations.     

The shape of random pattern-avoiding permutations

We initiate the study of limit shapes for random permutations avoiding a given pattern. Specifically, for patterns of length 3, we obtain delicate results on the asymptotics of distributions of positions of numbers in the permutations. We view the permutations as 0–1 matrices to describe the resulting asymptotics geometrically. We then apply our results to obtain a number of results on distributions of permutation statistics.     

A cycle lemma for permutation inversions

In this paper we study some properties of the inversion statistic. Some enumerative results concerning the permutations of the multiset {x₁^m₁,x₂^m₂} with respect to the inversion parameter are established and it is shown that these depend on gcd(m₁,m₂). Using a “cycle lemma”, a combinatorial proof of the results is given. Moreover, some applications to the Gaussian binomial coefficient are illustrated.     

On a conjecture about a class of permutation quadrinomials

Very recently, Tu et al. presented a sufficient condition on $(a_1,a_2,a_3)$, see Theorem 1.1, such that $f(x)=x^{3\cdot 2^m}+a_1{x}^{2^{m+1}+1}+a_2{x}^{2^m+2}+a_3{x}^3$ is a class of permutation polynomials over ${\mathbb{F}}_{2^n}$ with $n=2m$ and m odd. In this present paper, we prove that the sufficient condition is also necessary.     

A class of affinely equivalent Voronoi parallelohedra

Given any parallelohedron P, its affine class A (P), i.e., the set of all parallelohedra affinely equivalent to it, is considered. Does this affine class contain at least one Voronoi parallelohedron, i.e., a parallelohedron which is a Dirichlet domain for some lattice? This question, more commonly known as Voronoi’s conjecture, has remained unanswered for more than a hundred years. It is shown that, in the case where the subset of Voronoi parallelohedra in A (P) is nonempty, this subset is an orbifold, and its dimension (as a real manifold with singularities) is completely determined by its combinatorial type; namely, it is equal to the number of connected components of the so-called Venkov subgraph of the given parallelohedron. Nevertheless, the structure of this orbifold depends not only on the combinatorial properties of the parallelohedron but also on its affine properties.     

Substitution-closed pattern classes

The substitution closure of a pattern class is the class of all permutations obtained by repeated substitution. The principal pattern classes (those defined by a single restriction) whose substitution closure can be defined by a finite number of restrictions are classified by listing them as a set of explicit families.     

Elementary equivalent pairs of algebras associated with sets

The elementary equivalence of two full relation algebras, partition lattices or function monoids are shown to be equivalent to the second order equivalence of the cardinalities of the corresponding sets. This is shown to be related to elementary equivalence of permutation groups and ordinals. Infinite function monoids are shown to be ultrauniversal.Presented by Walter Taylor.The work of the second author was supported by a grant from the University of Cape Town Research Committee, and by the Topology Research Group from the University of Cape Town and the South African Council for Scientific and Industrial Research.     

A UNIFIED APPROACH TO A CLASS OF STERLING-TYPE PAIRS

Here presented is a unified approach to a wide class of symmetric Sfirling number pairs,which is determined by four complex parameters and includes as particular cases various previousextensions of Stirling numbers due to Carlicz, Howard, Koutras, Gould-Hopper, respectively.Certain Schlomilch-type formulas and congruence properties will be also exhibited.     

A generating functionology approach to a problem of Wilf

Wilf posed the following problem: determine asymptotically as n→∞ the probability that a randomly chosen part size in a randomly chosen composition of n has multiplicity m. One solution of this problem has been given by two of the authors 〈http://www.csc.ncsu.edu/faculty/savage/〉. In this paper, we study this question using the techniques of generating functions and singularity analysis.     

On a conjecture of Wilf

Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum