Related papers: Cycle index sum for non-k-equal configurations

Given two positive integers $n$ and $k$, we obtain a formula for the base size of the symmetric group of degree $n$ in its action on $k$-subsets. Then, we use this formula to compute explicitly the base size for each $n$ and for each $k\le…

We define a new basis of the algebra of quasi-symmetric functions by lifting the cycle-index polynomials of symmetric groups to noncommutative polynomials with coefficients in the algebra of free quasi-symmetric functions, and then…

In this paper we generalize the notion of the comparative index for the pair of Lagrangian subspaces which has fundamental applications in oscillation theory of symplectic difference systems and linear differential Hamiltonian systems. We…

We introduce ordered and unordered configuration spaces of 'clusters' of points in an Euclidean space $\mathbb{R}^d$, where points in each cluster satisfy a 'verticality' condition, depending on a decomposition $d=p+q$. We compute the…

Given a finite abelian group $G$, consider the complete graph on the set of all elements of $G$. Find a Hamiltonian cycle in this graph and for each pair of consecutive vertices along the cycle compute their sum. What are the smallest and…

In this note, we outline the general development of a theory of symmetric homology of algebras, an analog of cyclic homology where the cyclic groups are replaced by symmetric groups. This theory is developed using the framework of crossed…

We construct a Baum-Douglas type model for $K$-homology with coefficients in $\mathbb{Z}/k\mathbb{Z}$. The basic geometric object in a cycle is a $spin^c$ $\mathbb{Z}/k\mathbb{Z}$-manifold. The relationship between these cycles and the…

Symmetric homology is an analog of cyclic homology in which the cyclic groups are replaced by symmetric groups. The foundations for the theory of symmetric homology of algebras are developed in the context of crossed simplicial groups using…

We consider the group action of the automorphism group $\I_n=\aut(\Zz_n)$ on the set $\Zz_n$, that is the set of residue classes modulo $n$. Clearly, this group action provides a representation of $\I_n$ as a permutation group acting on $n$…

In this paper we present the construction of explicit quasi-isomorphisms that compute the cyclic homology and periodic cyclic homology of crossed-product algebras associated with (discrete) group actions. In the first part we deal with…

We consider a invariant Dirac operator D on a manifold with a proper and cocompact action of a discrete group G. It gives rise to an equivariant K-homology class [D]. We show how the index of the induced orbifold Dirac operator can be…

A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main example, we calculate the rational homology groups of spaces of even and odd maps $S^m \to S^M$,…

This paper defines and develops cycle indices for the finite classical groups. These tools are then applied to study properties of a random matrix chosen uniformly from one of these groups. Properties studied by this technique will include…

In this expository paper we present some ideas of algebraic topology (more precisely, of homology theory) in a language accessible to non-specialists in the area. A $1$-cycle in a graph is a set $C$ of edges such that every vertex is…

Consider a symplectic circle action on a closed symplectic manifold with non-empty isolated fixed points. Associated to each fixed point, there are well-defined non-zero integers, called weights. We prove that the action is Hamiltonian if…

In this paper we study the problem of determining the homology groups of a quotient of a topological space by an action of a group. The method is to represent the original topological space as a homotopy limit of a diagram, and then act…

We study the random walk on the symmetric group $S_n$ generated by the conjugacy class of cycles of length $k$. We show that the convergence to uniform measure of this walk has a cut-off in total variation distance after $\frac{n}{k} log n$…

This contains a new version of the so-called non-commutative Gauss algorithm for polycyclic groups. Its results allow to read off the order and the index of a subgroup in an (possibly infinite) polycyclic group.

We give a combinatorial description (including explicit differential-form bases) for the cohomology groups of the space of n distinct nonzero complex numbers, with coefficients in rank-one local systems which are of finite monodromy around…

We prove the cyclic sum formulas for certain two-parameter multiple series. These are new and non-trivial generalizations of the cyclic sum formulas for multiple zeta values and multiple zeta-star values.