Solomon is currently a student at the University of Pennsylvania where he is pursuing a Ph.D in Computer Science. He has a keen interest in all things Computer Science and Maths, so he decided to specialize in Programming Languages, an area which applies ideas from diverse areas of Computer Science, Mathematics and Philosophy. One of his favourite things is when connections are made between seemingly disparate areas of inquiry, so he has enjoyed reading the Graphical Linear Algebra blog for the connections between string diagrams and linear algebra. As the Lindström-Gessel-Vienot Lemma shows us, there is also a surprising combinatorial connection. Such occurrences make Solomon very happy!

Consider the familiar diagram for the matrix :

Suppose that we wanted to push some flow from the input nodes and to the output nodes and . Moreover, we want to do this in such a way that every output node receives some flow without any congestion in the network. More specifically, want to push flow from the input nodes to the output nodes in such a way that:

no two different paths that are part of the flow intersect at

a vertex (not even endpoints), and for each output node , there is a path from some input

node to .

For example, in the diagram above, we can push units of flow from to and units of flow from to .

We can also push units of flow from to and units of flow from to .

We are not allowed to push units of flow from to and units of flow from to , since then we would have two different paths intersecting at .

Also, we are not allowed to push only units of flow from to , since then there is no path from some to .

We call a flow that satisfies conditions (1) and (2) a viable flow. Now we will measure the signed weight a viable flow. The weight of a viable flow is simply the product of all weights on the edges that are part of the flow. To get the signed weight, we multiply the result by -1 if the flow links to . Otherwise, we multiply the result by +1.

For example, in our matrix example, the first viable flow has weight . The unsigned weight of this flow is since we push units of flow from to and units of flow from to . To get the signed weight of this flow, we multiply the unsigned weight by +1 since is linked to . A similar computation shows that the signed weight of the second viable flow is .

Interestingly, when we add signed weights of all the viable flows in the diagram, we get the determinant of the matrix associated with the diagram i.e. . Coincidence?

The Lindström-Gessel-Vienot Lemma (LGVL) tells us that this in fact is not a

coincidence. Specifically, LGVL associates to each finite directed acyclic

graph (dag) a matrix whose determinant is exactly equal to the

sum of the signed weights of the viable flows in . To make this relationship precise we shall give the statement of LGVL, but first we need to introduce some definitions and notation that we will need in order to give the statement of the theorem.

First, we introduce permutations. For any sets and we call any bijection a permutation from to . Now assume that sends to respectively. For example, let , and let send to respectively. Write in a row in that order, skip a line, then write below for each . Draw a line from to , and count the number of crossings in between the s and the s. Define the sign of be . For the permutation we have just described, there are six crossings, so the sign of the permutation is .

The sign of a permutation can also be computed by raising -1 to the

number of inversions of , where is the number of ordered pairs such that but . In the example above, the number of inversions of is 6 = , which gives the same result as before.

For the next few pieces of notation as well as the statement of LGVL, assume that is a finite directed acyclic graph. Assume further that has input nodes and output nodes .

As before, we are interested in viable flows, i.e. flows that have the property that:

no two different paths that are part of the flow intersect at

a vertex (not even endpoints), and for each output node , there is a path from some input

node to .

Each viable flow can described by an -tuple of paths , where is a path from the input node to the output node for some permutation . Let be this permutation.

For each directed path between any two vertices in , let be the product of the weights of the edges of the path. For any two vertices and , write for the sum

where ranges over all paths from to .

We are finally in a position to give the statement of LGVL, which is that





where ranges over all viable flows in !

LGVL holds for any finite directed acyclic graph, not just graphs that look like the example. For example consider the following graph:

or equivalently, the following string diagram:



The matrix for this is

.

It has determinant , and it is easy to check that each of these terms indeed gives a viable flow from the s to the s. Note also that the sign of each term is +1 since is sent to in each of the viable flows.

Now on the one hand, LGVL gives us a way of computing the sum of the signed weights of viable flows in a dag as the determinant of a certain matrix. On the other hand, given a matrix , we may compute the determinant of by doing the following. First, we find a special dag . Next, we find a set of input vertices and output vertices in . We check to see that . Then, we (easily!) compute the sum of the signed weights of viable flows in from to . Finally we conclude that the result is equal to the determinant of by LGVL. This compositional approach to computing the determinant often leads to useful and surprising results, as we shall see below.

Consider the determinant of the following matrix:

where denotes the greatest common divisor of and .

It is difficult to come up with a closed formula for this determinant by using say row-reductions or the co-factor expansion. However, using the strategy involving LGVL gives a simple and surprising result.

But first, a few more definitions! Given any positive integer , let denote the number of positive integers less than or equal to which are relatively prime to . For example, since the only numbers less than 6 that are relatively prime to 6 are 1 and 5. On the other hand since all the positive integers less than 7 are relatively prime to 1. In fact, it is easy to see that for any prime .

Also, for the purposes of the proof, we shall need to use the fact that

which is a property first established by Gauss. For example

.

Now we proceed to the actual computation. Consider the graph which has vertices: are the input vertices, are the output vertices and are intermediate vertices. If divides , put the edge with weight and the edge with weight 1.

What is the weight of all the paths from to ? Well each path from to can have only two edges: an edge of weight and an edge of weight 1, for some , where divides and divides . Of course the weight of this path is multiplied by 1, which is just . Therefore,





The last equality is justified by Gauss’ result.

By the LGVL, it follows that

where ranges over all viable flows from the input vertices to the output vertices .

Now the only path from to is since 1 has only one divisor, namely 1. As a result, for each prime , any path from to in a viable flow must pass through . This is because has only two divisors, namely 1 and , but the path through 1 has already been blocked by 1.

Consequently, for any product of two primes, any path from to must pass through . Repeating this argument for all the other numbers less than or equal to , we conclude that all the viable flows from to link to .

Now if there is a viable flow which links to for some , then there must be some such that is linked to with (why?). But this means that divides , a contradiction since ! So we must have , for all .

Thus, there is a unique viable flow from to i.e. the flow for which is sent to via . Thus, we conclude that the determinant of is , since the weight of each path in this flow is as we showed earlier: