2 00 5 An Introduction to q - Species

The combinatorial theory of species developed by Joyal provides a foundation for enumerative combinatorics of objects constructed from finite sets. In this paper we develop an analogous theory for the enumerative combinatorics of objects constructed from vector spaces over finite fields. Examples of these objects include subspaces, flags of subspaces, direct sum decompositions, and linear maps or matrices of various types. The unifying concept is that of a “q-species,” defined to be a functor from the category of finite dimensional vector spaces over a finite field with to the category of finite sets. 1 Definitions The combinatorial theory of species originated in the work of Joyal [7] in 1981 and has developed into a mature theory for understanding classical enumerative combinatorics and its generating functions [1]. More than thirty years ago Goldman and Rota [5, 6] began the systematic exploration of the “subset-subspace” analogy, the foremost example being the analogy between the binomial coefficients, which count subsets, and the q-binomial coefficients, which count subspaces. Their work has an interesting prehistory that is outlined in a short survey by Kung [10]. The aim of this paper is to further the subsetsubspace analogy with the development of the theory of species for structures associated to vector spaces over finite fields. This is not the first appearance of q-analogs nor the first use of vector spaces in the theory of species. Décoste [3, 4] defined canonical q-counting series by means of qsubstitutions in the cycle index series and in the asymmetry index series introduced by Labelle [11]. Also, Joyal [8] introduced the concept of a “tensorial species,” which is a functor from the category of finite sets (with bijections) to the category of vector spaces over a field of characteristic zero. However, the approach in this paper is different from the electronic journal of combinatorics 12(1) (2005), #R62 1 the earlier work. First of all, we are not q-counting ordinary combinatorial structures but counting structures associated to vector spaces over the field of order q. Second, a q-species is a functor from a category of vector spaces to the category of sets, whereas a tensorial species is a functor in the opposite direction. First we recall the definition of a combinatorial species. Let B be the category whose objects are finite sets and whose morphisms are bijections. A species is a functor F : B → B [7, 1]. For a finite set U , the set F [U ] is a collection of structures on the set U . Let Fq be the finite field of order q. Define Vq to be the category whose objects are finite dimensional vector spaces over Fq and whose morphisms are the linear isomorphisms. Definition 1.1. A q-species (or species of structures over Fq) is a functor F : Vq → B. Let F (N) q be the vector space of countable dimension whose elements are vectors (a1, a2, . . .) with a finite number of non-zero components. Let e1, e2, . . . be the standard basis and let En be the span of e1, . . . , en. Then E0 ⊂ E1 ⊂ · · · ⊂ En ⊂ En+1 ⊂ · · · is an increasing sequence of subspaces whose union is F (N) q . Let γn be the order of the general linear group GLn(q) of invertible n×n matrices over Fq and define γ0 = 1. Recall that γn = n ∏