Erratum to: “Solitary quotients of finite groups”

In this short note a correct proof of Theorem 3.3 from [T?rn?uceanu M., Solitary quotients of finite groups, Cent. Eur. J. Math., 2012, 10(2), 740–747] is given.     

A note on a characterization of generalized quaternion 2-groups

In this note, we answer an open problem posed in M. T?rn?ceanu (2010) [5], and obtain that the generalized quaternion 2-groups are the unique finite noncyclic groups whose posets of conjugacy classes of cyclic subgroups have breaking points.     

Counting lattice chains and Delannoy paths in higher dimensions

Lattice chains and Delannoy paths represent two different ways to progress through a lattice. We use elementary combinatorial arguments to derive new expressions for the number of chains and the number of Delannoy paths in a lattice of arbitrary finite dimension. Specifically, fix nonnegative integers n₁,…,n_d, and let L denote the lattice of points (a₁,…,a_d)∈Z^d that satisfy 0≤a_i≤n_i for 1≤i≤d. We prove that the number of chains in L is given by where . We also show that the number of Delannoy paths in L equals Setting n_i=n (for all i) in these expressions yields a new proof of a recent result of Duchi and Sulanke [9] relating the total number of chains to the central Delannoy numbers. We also give a novel derivation of the generating functions for these numbers in arbitrary dimension.     

Some properties of the Schröder numbers

In the paper, the authors present some properties, including convexity, complete monotonicity, product inequalities, and determinantal inequalities, of the large Schröder numbers and find three relations between the Schröder numbers and central Delannoy numbers. Moreover, the authors sketch generalizing the Schröder numbers and central Delannoy numbers and their generating functions.     

Eulerian quasisymmetric functions for the type B Coxeter group and other wreath product groups

Eulerian quasisymmetric functions were introduced by Shareshian and Wachs in order to obtain a q-analog of Euler?s exponential generating function formula for the Eulerian numbers (Shareshian and Wachs, 2010 [17]). They are defined via the symmetric group, and applying the stable and nonstable principal specializations yields formulas for joint distributions of permutation statistics. We consider the wreath product of the cyclic group with the symmetric group, also known as the group of colored permutations. We use this group to introduce colored Eulerian quasisymmetric functions, which are a generalization of Eulerian quasisymmetric functions. We derive a formula for the generating function of these colored Eulerian quasisymmetric functions, which reduces to a formula of Shareshian and Wachs for the Eulerian quasisymmetric functions. We show that applying the stable and nonstable principal specializations yields formulas for joint distributions of colored permutation statistics, which generalize the Shareshian–Wachs q-analog of Euler?s formula, formulas of Foata and Han, and a formula of Chow and Gessel.     

A note on lattice chains and Delannoy numbers

Fix nonnegative integers n₁,…,n_d and let L denote the lattice of integer points (a₁,…,a_d)∈Z^d satisfying 0?a_i?n_i for 1?i?d. Let L be partially ordered by the usual dominance ordering. In this paper we offer combinatorial derivations of a number of results concerning chains in L. In particular, the results obtained are established without recourse to generating functions or recurrence relations. We begin with an elementary derivation of the number of chains in L of a given size, from which one can deduce the classical expression for the total number of chains in L. Then we derive a second, alternative, expression for the total number of chains in L when d=2. Setting n₁=n₂ in this expression yields a new proof of a result of Stanley [Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, 1999] relating the total number of chains to the central Delannoy numbers. We also conjecture a generalization of Stanley's result to higher dimensions.     

Truncation Formulas for Invariant Polynomials of Matroids and Geometric Lattices

This paper considers the truncation of matroids and geometric lattices. It is shown that the truncated matroid of a representable matroid is again representable. Truncation formulas are given for the coboundary and M?bius polynomial of a geometric lattice and the spectrum polynomial of a matroid, generalizing the truncation formula of the rank generating polynomial of a matroid by Britz.     

A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants

A special case of Haiman?s identity [M. Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002) 371–407] for the character of the quotient ring of diagonal coinvariants under the diagonal action of the symmetric group yields a formula for the bigraded Hilbert series as a sum of rational functions in q,t. In this paper we show how a summation identity of Garsia and Zabrocki for Macdonald polynomial Pieri coefficients can be used to transform Haiman?s formula for the Hilbert series into an explicit polynomial in q,t with integer coefficients. We also provide an equivalent formula for the Hilbert series as the constant term in a multivariate Laurent series.     

Finite non-cyclic p -groups whose number of subgroups is minimal

Recent results of Qu and Tărnăuceanu explicitly describe the finite p-groups which are not elementary Abelian and have the property that the number of their subgroups is maximal among p-groups of a given order. We complement these results from the bottom level up by completely determining the non-cyclic finite p-groups whose number of subgroups among p-groups of a given order is minimal.     

有限循环群的Fuzzy子群的等价类数

有限循环群G的F子群可以有无数个．但是．若当两个F子群的水平集构成的集合相等就称其等价的话，那么其等价类数是有限的。通过研究群的合成群列、商群列以及数的因数列和极大因数列找出了有限循环群的极大F子群和F子群的等价类数的求解公式．并给出二者之间的关系式．     

Counting Lattice Paths With Four Types of Steps

The main purpose of this paper is to derive generating functions for the numbers of lattice paths running from (0, 0) to any (n, k) in \({\mathbb{Z} \times \mathbb{N}}\) consisting of four types of steps: horizontal H = (1, 0), vertical V = (0, 1), diagonal D = (1, 1), and sloping L = (–1, 1). These paths generalize the well-known Delannoy paths which consist of steps H, V, and D. Several restrictions are considered. However, we mainly treat with those which will be needed to get the generating function for the numbers R(n, k) of these lattice paths whose points lie in the integer rectangle \({\{(x, y) \in \mathbb{N}^2 : 0 \leq x \leq n, 0 \leq y \leq k\}}\) . Recurrence relation, generating functions and explicit formulas are given. We show that most of considered numbers define Riordan arrays.     

有限Abel群的Fuzzy子群的等价分类及其类数

定义Fuzzy子群的两种等价关系,给出了有限Abel群的Fuzzy子群在这两种等价关系下的等价类数的求解公式。     

On Connes’ trace formula for the Hankel transformation of order ?1/2

A local Hankel transformation of order ?1/2 is defined for every finite place of the field of rational numbers. Its inversion formula and the Plancherel type theorem are obtained. A Connes type trace formula is given for each local Hankel transformation of order ?1/2. An S-local Connes type trace formula is derived for the S-local Hankel transformation of order ?1/2. These formulas are generalizations of Connes?? corresponding trace formulas in 1999.     

Primitive Idempotents of Schur Rings

In this paper, we explore the nature of central idempotents of Schur rings over finite groups. We introduce the concept of a lattice Schur ring and explore properties of these kinds of Schur rings. In particular, the primitive, central idempotents of lattice Schur rings are completely determined. For a general Schur ring S, S contains a maximal lattice Schur ring, whose central, primitive idempotents form a system of pairwise orthogonal, central idempotents in S. We show that if S is a Schur ring with rational coefficients over a cyclic group, then these idempotents are always primitive and are spanned by the normal subgroups contained in S. Furthermore, a Wedderburn decomposition of Schur rings over cyclic groups is given. Some examples of Schur rings over non-cyclic groups will also be explored.     

On the lattice automorphisms of the finite Chevalley groups

A lattice automorphism of a group is defined to be an automorphism of its lattice of subgroups. For a large class of finite simple Chevalley groups, it is shown that every lattice automorphism is induced by a group automorphism. However, this does not hold for all finite simple Chevalley groups G, as is shown by explicit construction in the case G=PSL(3, q).     

On the spectrum and the geometry of a spherical space form II

In this paper certain relations between the numerical coefficients of the Poisson formula of Part I and geometrical data of a spherical space form M, dim M = 2m+1 are investigated. The results yield an explicit relation between the spectrum of M and the Poincaré map of certain closed geodesics of M. Furthermore, explicit formulas for the multiplicities of the eigenvalues of the Laplacian of M are derived by means of the Poisson formula. At the end of the paper the information about M is examined which is contained in a finite part of spec(M). A partial answer is given in the Corollaries 3 and 6.     

Euler Characteristics on virtually free products

We define Euler characteristics on classes of residually finite and virtually torsion free groups and we show that they satisfy certain formulas in the case of amalgamated free products and HNN extensions over finite subgroups. These formulas are obtained from a general result which applies to the rank gradient and the first L²?Betti number of a finitely generated group.     

A formula for the Whitehead group of a three-dimensional crystallographic group

We give an explicit formula for the Whitehead group of a three-dimensional crystallographic group Γ in terms of the Whitehead groups of the virtually infinite cyclic subgroups of Γ.     

On formulas for Dedekind sums and the number of lattice points in tetrahedra

This paper explores a simple yet powerful relationship between the problem of counting lattice points and the computation of Dedekind sums. We begin by constructing and proving a sharp upper estimate for the number of lattice points in tetrahedra with some irrational coordinates for the vertices. Besides providing a sharper estimate, this upper bound (Theorem 1.1) becomes an equality (i.e. gives the exact number of lattice points) in a tetrahedron where the lengths of the edges divide each other. This equality condition can then be applied to the explicit computation of the classical Dedekind sums, a topic that is the central focus in the second half of our paper. In this half of the paper, we come up with a number of interesting results related to Dedekind sums, based on our upper estimate (Theorem 1.1). Among these findings, Theorem 1.9 and Theorem 1.10 deserve special attention, for they successfully generalize two of Apostol's formulas in [T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1997], and also directly imply the famous Reciprocity Law of Dedekind sums.     

A Note on Counting Homomorphisms of Paths

We obtain two identities and an explicit formula for the number of homomorphisms of a finite path into a finite path. For the number of endomorphisms of a finite path these give over-count and under-count identities yielding the closed-form formulae of Myers. We also derive finite Laurent series as generating functions which count homomorphisms of a finite path into any path, finite or infinite.