algebra and enumerative combinatorics We learn something when we can interpret combinatorial numbers as orders of algebraic structures, especially groups. Lagrange’s Theorem means that group orders tend to have a rich arithmetic structure. I first learned from Norman’s paper on chip-firing games the fact that the number of spanning trees of a graph is the order of an abelian group canonically associated with the graph (a reduced homology group). For the complete graph Kn, this group is the direct product of n− 2 copies of the cylic group of order n. Endomorphisms and automorphisms Rather than an order, we can impose algebraic structure on the set X, and ask for maps which are endomorphisms, or partial maps which are partial isomorphisms. This is not well explored yet, but here is a curiosity. Theorem The number of partial isomorphisms of rank k of an n-dimensional vector space over a field of order q is equal to the number of endomorphisms of rank k of the same vector space. Corollary The order of the inverse semigroup of partial isomorphisms of an n-dimensional vector space of rank n over a field of order q is equal to the order of the semigroup of endomorphisms, namely qn 2 .
Download Full PDF Version (Non-Commercial Use)