20 research outputs found
Omitting parentheses from the cyclic notation
The purpose of this article is to initiate a combinatorial study of the
Bruhat-Chevalley ordering on certain sets of permutations obtained by omitting
the parentheses from their standard cyclic notation. In particular, we show
that these sets form a bounded, graded, unimodal, rank-symmetric and
EL-shellable posets. Moreover, we determine the homotopy types of the
associated order complexes.Comment: new results adde
On the definition of linear independence
We discuss a certain very common flaw in the definition of linear independence, which is one of the most important concepts taught in any college or university course of Linear Algebra. This note may be useful to lecturers and students which teach and study Linear Algebra of any level and like the mathematically rigorous approach
Structure of the Group of Balanced Labelings on Graphs, its Subgroups and Quotient Groups
We discuss functions from edges and vertices of an undirected graph to an
Abelian group. Such functions, when the sum of their values along any cycle is
zero, are called balanced labelings. The set of balanced labelings forms an
Abelian group. We study the structure of this group and the structure of two
closely related to it groups: the subgroup of balanced labelings which consists
of functions vanishing on vertices and the corresponding factor-group. This
work is completely self-contained, except the algorithm for obtaining the
3-edge-connected components of an undirected graph, for which we make
appropriate references to the literature.Comment: 22 page