One of my favorite results in algebraic combinatorics is a surprisingly useful lemma which allows a combinatorial interpretation of the determinant of certain integer matrices. One of its more popular uses is to prove an equivalence between three other definitions of the Schur functions (none of which I have given yet), but I find its other applications equally endearing.
Let be a locally finite directed acyclic graph, i.e. it has a not necessarily finite vertex set with finitely many edges between each pair of vertices such that no collection of edges forms a cycle. For example, could be with edges and , which we’ll denote the acyclic plane. Assign a weight to each edge and assign to a path the product of the weights of its edges. Given two vertices let denote the sum of the weights of the paths from to . Hence even if there are infinitely many such paths this sum is well-defined formally, and if there are only finitely many paths between two vertices then setting each weight to gives a well-defined non-negative integer.
Let and be a collection of vertices called sources and vertices called sinks. We are interested in -tuples of paths, hereafter to be referred to as -paths, sending each source to a distinct sink. Let be the matrix such that . Then the permanent of counts the number of -paths, but this is not interesting as permanents are hard to compute.
A -path is called non-intersecting if none of the paths that make it up share a vertex; in particular, each is sent to distinct . A non-intersecting path determines a permutation of the vertices; let the sign of a non-intersecting -path be the sign of this permutation.
Lemma (Lindström, Gessel-Viennot): is the signed sum of the weights of all non-intersecting -paths.
Corollary: If the only possible permutation is (i.e. is non-permutable), then is the sum of the weights of all non-intersecting -paths.
The idea of the proof is straightforward: one defines a sign-reversing (weight-preserving) involution on -paths so that in the signed sum only the fixed points survive and the others cancel. (This is more or less equivalent to a use of inclusion-exclusion, but in many cases in combinatorics it is more natural to define the involution directly.) The involution is roughly as follows: if a path is non-intersecting, fix it (of course). Otherwise, there is some vertex that two paths share as a vertex. Flip the part of the paths after , which changes the sign of the induced permutation.
The only difficulty is to make sure this is really an involution, i.e. that the same portion of path gets flipped if one applies the map twice. To ensure this, let be the smallest index such that the path from intersects another path and let be the largest index of a path which intersects the above path. These indices remain unchanged after the involution, so the vertex also remains unchanged. If you want the full details, there is a nice introduction by Benjamin and Cameron in the AMM, Vol. 112, No. 6. Benjamin and Cameron describe a nice application to the evaluation of a determinant of Catalan numbers, so I won’t describe it; it makes more sense with the diagram anyway.
Gessel and Viennot’s original motivation was precisely an application to the acyclic plane, and indeed the lemma is sometimes stated for the plane only. Let and be strictly increasing sequences of non-negative integers and define sources and sinks . The number of paths from to is readily verified to be , so is the binomial determinant
Apparently these determinants have something to do with Chern classes. When either the or the are consecutive integers, this binomial determinant has a product formula (for example, see this AoPS thread), which Gessel and Viennot prove combinatorially using their lemma.
The lemma immediately implies that determinants are multiplicative. Actually, it implies an even stronger result, the Cauchy-Binet formula. Consider now three sets of vertices and construct two matrices , one for paths and the other for paths . Note that are not necessarily square, but is a square matrix counting paths which pass through some -vertices. Then counts nonintersecting paths which must therefore pass through distinct -vertices. If , there are no such paths. Otherwise, for every subset of of size , every -path factors as a product of -paths , and the signs multiply in the expected way. Letting denote the matrices counting such paths, it follows that
As a special case, if , then we obtain the identity
This identity implies that the square of the volume of a parallelpiped in with vertices given by the entries of is equal to the sum of the squares of the volumes of its projections to the standard copies of obtained by ignoring some of the coordinates. This is a high-dimensional generalization of the Pythagorean theorem, and it can also be used to prove the matrix-tree theorem. In fact, since the minors of a matrix to which Gessel-Viennot applies have almost exactly the same combinatorial interpretation as the full determinant, we even know that matrices of the form are positive-semidefinite. (More generally, this is true for any matrix coming from a nonpermutable graph.)
The lemma implies that non-intersecting random walks are a determinantal process, which connects them to many other mysterious processes. I wish I knew what to make of this.
Over the summer I conjectured that the lemma could be used to prove the identity
for the adjacency matrix of a finite graph . My idea was that one could construct a “blowup” of with the property that Gessel-Viennot applied, and Joel Lewis and I tried a few constructions that looked something like this: for each non-negative integer , has vertices where is a vertex of . For each edge of , has edges of weight . Finally, add the vertices and edges for each finite of weight .
Now it follows that a walk from to is precisely a walk of some length from to on of weight , hence is precisely equal to the entry of . So let the vertices be sources and the vertices be sinks.
Unfortunately, as constructed is not non-permutable. It’s possible one might be able to define a second sign-reversing involution on non-intersecting -paths in this setup, but I would really like it if could be modified to be nonpermutable. In all honesty I haven’t given this problem much thought since the summer, so it could be that the proof is relatively straightforward from here.