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 a...

Let Q be a finite set and let k be a non negative integer. A (partial) function f: Q + Q is a k-map if ]sf’] < k, Vq E Q. A transformation semigroupx = (Q, S) is a k-t.s. if each s ES is a k-map. Let TS be the collection of all (finite) transformation semigroups and let c: TS + N be the complexity function. For background on complexity see [2]. The main theorem of this paper shows that if X is ...

In collaboration with João Araújo and others, I have been thinking about relations between permutation groups and transformation semigroups. In particular, what properties of a permutation group G on a set Ω guarantee that, if f is any non-permutation on Ω (or perhaps any non-permutation of rank k), then the transformation semigroup 〈G, f〉 has specified properties. There is far too much to summ...

We prove that any countable set of surjective functions on an infinite set of cardinality אn with n ∈ N can be generated by at most n/2 + 9n/2 + 7 surjective functions of the same set; and there exist n/2 + 9n/2 + 7 surjective functions that cannot be generated by any smaller number of surjections. We also present several analogous results for other classical infinite transformation semigroups ...