A Primer: Generating Functions: Part II: for RMO/INMO 2019

We shall now complicate the situation a little bit. Let us ask for the combinations of the symbols $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ with repetitions of each symbol allowed once more in the combinations. For example, let there be only two symbols $\alpha_{1}, \alpha_{2}$ . Let us look for combinations of the form:

$\alpha_{1}$ , $\alpha_{2}$ , $\alpha_{1}\alpha_{2}$ , $\alpha_{1}\alpha_{1}$ , $\alpha_{2}\alpha_{2}$ , $\alpha_{1}\alpha_{1}\alpha_{2}$ , $\alpha_{1}\alpha_{2}\alpha_{2}$ , $\alpha_{1}\alpha_{1}\alpha_{2}\alpha_{2}$

where, in each combination, each symbol may occur once, twice, or not at all. The OGF for this can be constructed by reasoning as follows: the choices for $\alpha_{1}$ are not- $\alpha_{1}$ , $\alpha_{1}$ once, $\alpha_{1}$ twice. This is represented by the factor $(1+\alpha_{1}t+\alpha_{1}^{2}t^{2})$ . Similarly, the possible choices for $\alpha_{2}$ correspond to the factor $(1+\alpha_{2}t+\alpha_{2}^{2}t^{2})$ . So, the required OGF is $(1+\alpha_{1}t+\alpha_{1}^{2}t)(1+\alpha_{2}t+\alpha_{2}^{2}t^{2})$

On expansion, this gives : $1+(\alpha_{1}+\alpha_{2})t+(\alpha_{1}\alpha_{2}+\alpha_{1}^{2}+\alpha_{2}^{2})t^{2}+(\alpha_{1}^{2}\alpha_{2}+\alpha_{1}\alpha_{2}^{2})t^{3}+(\alpha_{1}^{2}\alpha_{2}^{2})t^{4}$

Note that if we omit the term 1 (which corresponds to not choosing any $\alpha$ ), the other 8 terms correspond to the 8 different combinations listed in (*). Also, observe that the exponent r of the $t^{r}$ tells us that the coefficient of $t^{r}$ has the list or inventory of the r-combinations (under the required specification — in this case, with the restriction on repetitions of symbols) in it:

$\bf{Illustration}$

In the light of the foregoing discussion, let us now take up the following question again: in how many ways, can a total of 16 be obtained by rolling 4 dice once?; the contribution of each die to the total is either a “1” or a “2” or a “3” or a “4” or a “5” or a “6”. The contributions from each of the 4 dice have to be added to get the total — in this case, 16. So, if we write: $t^{1}+t^{2}+t^{3}+t^{4}+t^{5}+t^{6}$

as the factor corresponding to the first die, the factors corresponding to the other three dice are exactly the same. The product of these factors would be:

(*) $(t+t^{2}+t^{3}+t^{4}+t^{5}+t^{6})^{4}$

Each term in the expansion of this would be a power of t, and the exponent k of such a term $t^{k}$ is nothing but the total of the four contributions which went into it. The number of times a term $t^{k}$ can be obtained is exactly the number of times k can be obtained as a total on a throw of the four dice. So, if $\alpha_{k}$ is the coefficient of $t^{k}$ in the expansion, $\alpha_{16}$ is the answer for the above question. Further, since (*) simplifies to $(\frac{t(1-t^{6})}{1-t})^{4}$ , it follows that the answer for the above question tallies with the coefficient specified in the following next question: calculate the coefficient of $t^{12}$ in $(\frac{(1-t^{6})}{(1-t)})^{4}$ .6

Now, consider the following problem: Express the number $N(n,p)$ of ways of obtaining a total of n by rolling p dice, as a certain coefficient in a suitable product of binomial expansions in powers of t. [ this in turn, is related to the observation that the number of ways a total of 16 can be obtained by rolling 4 dice once is the same as the coefficient of $t^{12}$ in $(\frac{1-t^{6}}{1-t})^{4}$ ]:

So, we get that $N(n,p)=$ coefficient of $t^{n-p}$ in $(\frac{1-t^{6}}{1-t})^{p}$

Let us take an example from a graphical enumeration:

A $\it {graph}$ $G=G(V,F)$ is a set V of vertices a, b, c, …, together with a set $E=V \times V$ of $\it {edges}$ $(a,b), (a,a), (b,a), (c,b), \ldots$ If $(x,y)$ is considered the same as $(y,x)$ , we say the graph is $\it{undirected}$ . Otherwise, the graph is said to be $\it{directed}$ , and we say ‘ $(a,b)$ has a direction from a to b’. The edge $(x,x)$ is called a loop. The graph is said to be of order $|V|$ .

If the edge-set E is allowed to be a multiset, that is, if an edge $(a,b)$ is allowed to occur more than once, (and, this may be called a ‘multiple edge’), we refer to the graph as a general graph.

If $\phi_{5}(n)$ and $\psi_{5}(n)$ denote the numbers of undirected (respectively, directed) loopless graphs of order 5, with n edges, none of them a multiple edge, find the series $\sum \phi_{5}(n)t^{n}$ and $\sum \psi_{5}(n)t^{n}$ .

Applying our recently developed techniques to the above question, a graph of 5 specified vertices is uniquely determined once you specify which pairs of vertices are ‘joined’. Suppose we are required to consider only graphs with 4 edges. This would need four pairs of vertices to be selected out of the total of $5 \choose 2$ equal to 10 pairs that are available. So selection of pairs of vertices could be made in $10 \choose 4$ ways. Each such selection corresponds to one unique graph, with the selected pairs being considered as edges. More informally, having selected a certain pairs of vertices, imagine that the vertices are represented by dots in a diagram and join the vertices of each selected pair by a running line. Then, the “graph” becomes a “visible” object. Note that the number of graphs is just the number of selections of pairs of vertices. Hence, $\phi_{5}(4)=10 \choose 4$ .

Or, one could approach this problem in a different way. Imagine that you have a complete graph on 5 vertices — the “completeness” here means that every possible pair of vertices has been joined by an edge. From the complete graph which has 10 edges, one has to choose 4 edges — any four, for that matter — in order to get a graph as required by the problem.

On the same lines for a directed graph, one has a universe of 10 by 2, that is, 29 edges to choose from, for, each pair x,y gives rise to two possible edges $(x,y)$ and $(y,x)$ . Hence,

$\psi_{5}(4)=20 \choose 4$ .

Thus, the counting series for labelled graphs on 5 vertices is $1 + \sum_{p=1}^{10} {10 \choose p}t^{p}$
and the counting series for directed labelled graphs on 5 vertices is
$1+ \sum_{p=1}^{20}{20 \choose p}t^{p}$ .

Finally, the OGF for increasing words on an alphabet ${a,b,c,d,e}$ with $a<b<c<d<e$ is

$(1+at+a^{2}t^{2}+\ldots)(1+bt+b^{2}t^{2}+\ldots)(1+ct+c^{2}t^{2}+\ldots)\times (1+dt+d^{2}t^{2}+\ldots)(1+et+e^{2}t^{2}+\ldots)$

The corresponding OE is $(1+t+t^{2}+t^{3}+\ldots)^{5}$ which is nothing but $(1-t)^{-5}$ (this explains the following problem: Verify that the number of increasing words of length 10 out of the alphabet $\{a,b,c,d,e \}$ with $a<b<c<d<e$ is the coefficient of $t^{10}$ in $(1-t)^{-5}$ ).