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I plan to take an undergraduate combinatorics course at my university, and I've seen generating functions as part of the curriculum. I am not a math student but I've seen some pure math courses outside the realm of calculus. My calculus knowledge has partially dissipated over time. I would like to know how much should I know about power series. Is knowledge about summations properties sufficient? I'll appreciate any other helpful advice.

RobPratt
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Mangostino
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  • Depends on the combinatorics course. Basic combinatorics essentially involves sums and products. Power series link more directly to functional analysis and linear algebra than they do to combinatorics, depending on the presentation – kevinkayaks Jun 30 '21 at 17:10
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    https://www2.math.upenn.edu/~wilf/gfology2.pdf is one of the most popular books on generating functions, and also pretty fun. I suggest skimming this to get an idea of what you'll likely encounter. – Merosity Jun 30 '21 at 18:04
  • Thank you for that reference! Already answered some crucial doubts I had. – Mangostino Jun 30 '21 at 18:12
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    What do you mean by "understand"? As for gaining intuition, some calculus experience will be very helpful, particularly around infinite series. Wilf's book is helpful here, as mentioned by Merosity. As for doing the proofs rigorously, you will need to have some abstract algebra experience (or the patience required to do everything by hand...) and some "sequences and series" type reasoning (but no actual real analysis). Chapter 3 of my Algebraic Combinatorics notes has many of the details. – darij grinberg Jun 30 '21 at 19:31
  • You might get some ideas from the answers to this question: https://math.stackexchange.com/questions/3142386/how-can-i-learn-about-generating-functions/3142387 – awkward Jul 01 '21 at 13:21

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Maybe knowing Newton's binomial theorem can help.

The binomial theorem that you learn in high school says $$ (x+y)^n = \sum_{k=0}^n \binom n k x^k y^{n-k} $$ for $n\in\{0,1,2,3,\ldots\}.$ Here we have \begin{align} \binom n k = {} & \frac{n!}{k!(n-k)!} \tag 1 \\[8pt] = {} & \frac{n(n-1)(n-2)\cdots(n-k+1)}{k!}. \tag 2 \end{align} Both lines $(1)$ and $(2)$ are correct, but line $(2)$ makes sense if $n = 3.14159$ or $n=-3.$ And in particular, if $n$ is a positive integer that is less than $k,$ then $\binom n k=0,$ so that we can write $$ (x+y)^n = \sum_{k=0}^\infty \binom n k x^k y^{n-k} $$ (i.e. with $\sum_{k=0}^\infty$ in place of $\sum_{k=0}^n$) provided $|y/x|<1.$ If $|y/x|\ge 1$ then the series does not converge.

Now notice that in combinatorics you will work with infinite series like this without worrying about whether they converge or not.

A series like this is used in proving that "negative sets are multisets." Multisets of size $5$ of a set $\{a,b,c\}$ of size $3$ are: $$ \{a,a,a,a,a\}, \{a,a,a,a,b\}, \{a,a,b,c,c\}, \text{ etc.} $$ The number of multisets of size $5$ of a set of size $3$ is $$ \left|\binom {-3}5\right| = \left| \frac{(-3)(-4)(-5)(-6)(-7)}{5\cdot4\cdot3\cdot2\cdot1} \right| = 21. $$ Recall that with $n\ge0$ and $n$ an integer, we have, for example $$ \binom 8 3 = \text{the number of subsets of size 3 of a set of size 8.} $$

Michael Hardy
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  • So, when working with infinite series I should worry more about how to manipulate them algebraically. Is that correct? – Mangostino Jun 30 '21 at 19:30
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    @Mangostino : Yes. Generating functions in combinatorics are "formal" power series, i.e. power series treated as algebraic objects without any questions of convergence. – Michael Hardy Jun 30 '21 at 21:15