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It is well known that the terms of the power series of exponential and trigonometric functions often involve the factorial function, essentially as a consequence of iterating the power rule.

My question is whether there is another way to view the occurrence of factorials so often; factorials are often involved in counting/combinatorics, so is there a combinatorial interpretation of why this happens, or some other interesting interpretation?

More generally, is there any way I can look at the power series of $e^x$ or some other function and intuitively understand why factorials are involved, rather than just thinking of the power rule and derivatives?

Beren Gunsolus
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  • The binomial theorem for $n\in N$ can be stated as $(1+x)^n=1 +x n + x^2n(n-1)/2!+...$ which can be proven combinatorially. Isaac Newton found this held, as an infinite series,for any real $n$, provided $|x|<1.$ Probably by, for example, squaring the series with $n=1/2$ and,after simplifying , getting $1+x$. – DanielWainfleet Dec 02 '15 at 02:48
  • Something kinda helpful here is verifying exp(x)*exp(y)=exp(x+y) using the power series definition: we get combinatorics from the binomial theorem, and the fact that binomial coefficients are a ratio of factorials is key to the equality holding, which sort of helps understand why we expect something 'combinatorial' in the coefficients of exp. – Beren Gunsolus Mar 12 '23 at 03:47

4 Answers4

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Look at chapter 2 of "generatingfunctionology", available here: https://www.math.upenn.edu/~wilf/DownldGF.html

There is a discussion of ordinary generating functions and exponential generating functions, the ones with factorials.

Wilf says "We remarked earlier that the convolution of sequences that is shown in (2.3.4) is useful in counting problems where structures of size n are obtained by stitching together structures of sizes r and n − r in all possible ways. Correspondingly, the convolution (2.3.3) is useful in combinatorial situations where we not only stitch together two such structures, but we also relabel the structures. For then, roughly speaking, there are $\binom{n}{r}$ ways to choose the new labels of the elements of the structure of size r, as well as $a_r$ ways to choose that structure and $b_{n−r}$ ways to choose the other one."

marty cohen
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We have $\sin^2t+\cos^2t=1$ and $\cosh^2t-\sinh^2t=1$, both of which are born of the implicit algebraic equations of the circle and hyperbola, $x^2\pm y^2=r^2$. In other words, these two geometric shapes, studied since the time of the ancient Greeks, are defined by constant or bounded sums of powers. And where are factorials known to occur ? In the expression of binomial coefficients, which famously characterize the binomial theorem, which expands the power of a sum into a sum of powers. So the intrinsic umbilical link between trigonometric or hyperbolic functions and factorials or binomial coefficients is as natural and intuitive as can be. Thus, it should come as no surprise that the beta and $\Gamma$ functions of argument $1/n$ are inherently connected to geometric figures described by equations of the form $x^n+y^n=r^n$, yielding, for instance, the celebrated identities $\Gamma\bigg(\dfrac12\bigg)=\sqrt\pi$ and $B\bigg(\dfrac12~,~\dfrac12\bigg)=\pi$.

Lucian
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  • It doesn't come as a surprise, but the examples you give are just that: examples. The mystery seems even more mysterious now. May I put the question more concretely? Factorials imply orderings; division by a factorial implies equivalence of orderings. But what are the things being reordered, and why are they equivalent? – senderle Mar 17 '17 at 00:03
  • @senderle: Are you asking me what $(-1/2)!=\sqrt\pi$ or $\displaystyle{0\choose+1/2}={0\choose-1/2}=2/\pi$ are supposed to order ? Let me just put it this way: factorials and binomial coefficients of integer arguments imply orderings. Just like the phrase ‘a person posting comments on a website’ does not and cannot accurately capture or describe what a human being actually is, so can a mathematical object not be reduced to certain of its properties, such as the ones that originally led to its initial discovery, for instance. Indeed, such views are rather restrictive of its full potential. – Lucian Mar 17 '17 at 03:35
  • I disagree. 1) There's no reason arguments have to be integers for factorials to imply orderings. They could imply orderings of sequences of non-integers, or of sequences of operations on non-integers. 2) I think questions like the one I'm asking are sources of novel interpretations and new insights. It's not as if I'm claiming this is the only way to interpret factorial coefficients. – senderle Mar 17 '17 at 11:49
  • @senderle: Perhaps we are talking about different things. – Lucian Mar 17 '17 at 12:26
  • Perhaps so. Parts of this answer are suggestive along the lines I'm thinking, if that helps. But even there, I feel like there's a lot being left unsaid. – senderle Mar 17 '17 at 14:41
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Many classes of functions can be developed into a function series. There are different kinds of function series possible. Each analytic function e.g. can locally be given by a convergent power series. A simple form of such a power series is the Taylor series. A Taylor series has some nice properties.

As you've already noted, the denominator of the Taylor coefficient is already a factorial. This comes from the $n$-th derivative of the summand term $x^{n}$ and is simply a result of the properties of the Taylor series. And it is a result of the power rule of differentiation. Higher product rule and higher power rule contain partitions of derivatives of lower degree. This is the combinatorial nature of the factorial in Taylor series and in derivatives.

The numerator of the $n$-th Taylor coefficient, the $n$-th local derivative, can also be interpreted combinatorially. It can be described as Bell polynomial. That is a partition polynomial. It contains all partitions of all derivatives of lower degree, and its coefficients contain the number of integer partitions and the number of set partitions; they are called multinomial coefficients of the third kind. Therefore this coefficients contain factorials in the numerator and in the denominator. Speaking pictorially, partial derivatives and higher derivatives are integer partitions. And this partition structure is maintained in exponential generating functions and in ordinary generating functions.

Factorials are the building blocks of the frequency numbers of combinatorial objects, i.e. of combinatorial numbers. They form binomials, multinomials and other combinatorial numbers which express the frequency of combinatorial structures like combinations, permutations, integer partitions, set partitions and combinatorial objects which are assembled from this basic structures.

The combinatorial interpretation and the interpretation by functions are often two pictures of the same thing. Both are interrelated by generating functions.

IV_
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To take a different starting point, you could think about the formula:

$$ e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n $$

which falls out from thoughts of compound interest compounded infinitely often.

Rob Arthan
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