An extensive theory of the topology associated with finite groups of continuous transformations remains to be developed. Mr. M. Richardson has evaluated the Betti numbers of the "domain of discontinuity" in a case of considerable generality and his methods' yield equally well certain modulo p invariants. They seem to be inadequate, however, when p equals a, the order of the group. We shall brie...

We derive limit laws for random combinatorial structures using singularity analysis of generating functions. We begin with a study of the Boltzmann samplers of Flajolet and collaborators, a useful method for generating large discrete structures at random which is useful both for providing intuition and conjecture and as a possible proof technique. We then apply generating functions and Boltzman...

We show that a non-hyperbolic orthogonal involution on a central simple algebra over a field of characteristic 6= 2 remains non-hyperbolic over some splitting field of the algebra.

Define I n(α) to be the set of involutions of {1, 2, . . . , n} with exactly k fixed points which avoid the pattern α ∈ Si, for some i ≥ 2, and define I n(∅;α) to be the set of involutions of {1, 2, . . . , n} with exactly k fixed points which contain the pattern α ∈ Si, for some i ≥ 2, exactly once. Let in(α) be the number of elements in I k n(α) and let i k n(∅;α) be the number of elements in...

We present a general construction of involutions on integer partitions which enable us to prove a number of modulo 2 partition congruences. Introduction The theory of partitions is a beautiful subject introduced by Euler over 250 years ago and is still under intense development [2]. Arguably, a turning point in its history was the invention of the “constructive partition theory” symbolized by F...

Subbarao and Andrews have observed that the combinatorial technique used by F. Franklin to prove Eulers famous partition identity (l-x)(l-x)(l-x)(l-x*) ••• = 1-x-x +x +x -x -x + ••• can be applied to prove the more general formula l-x-xy(l-xy) -xy(±-xy)(±-xy) xy (1 xy) (1 xy) (1 xy) = 1 -x-xy+xy+xy -xy -xy + • •• which reduces to Eulers when y = 1. This note shows that several finite versions o...

A permutomino of size n is a polyomino determined by particular pairs (π1, π2) of permutations of length n, such that π1(i) 6= π2(i), for 1 ≤ i ≤ n. In this paper we consider the class of convex permutominoes which are symmetric with respect to the diagonal x = y. We determine the number of these permutominoes according to the dimension and we characterize the class of permutations associated t...

A q-Lagrange inversion theorem due to A. M. Garsia is proved by means of two sign-reversing, weight-preserving involutions on Catalan trees.

The reduced anisotropic unitary Whitehead groups of henselian division algebras with involutions are computed in the cases where centers residue special types.

The notion of an asymptotic bijection is introduced and used to give bijective proofs of infinite summation formulas for set partitions (Dobinski's formula) and involutions.