Numerator of harmonic number?

Question

I have a harmonic number that is the sum of $z$ terms as follows:

$$1+\frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+\ldots+ \frac{1}{z}$$

$$=\frac{(2)(3)(4)\ldots(z-1)(z)+ (1)(3)(4)\ldots(z-1)(z)+ (1)(2)(4)\ldots(z-1)(z) +\ldots+ (1)(2)(3)(4)\ldots(z-1)}{z!}$$

The denominator is easy to describe and notate, it’s just the factorial of the number of terms in the harmonic sum.

The numerator is the sum of the products of all distinct combinations of $z-1$ distinct integers from the set of integers $\{1,2,3,4,...,z-1,z\}$... but it’s quite complex to work with, and even the description is complicated.

Is there a mathematical notation that expresses a number of this kind? What’s it called? I’ve tried searching through arrivals about combinatorics, harmonic analysis, binomial theorem... but I haven’t got anywhere.

Could someone please point me in the right direction?

Answer 1

You describe the numerator as follows:

The sum of the product of all distinct combinations of $z-1$ distinct integers from the set of integers $\{1,2,3,\ldots,z\}$.

Luckily, there’s a way to notate an even more general sum, the sum of all distinct combinations of $k$ distinct numbers from a given multiset $\{x_1,x_2,\ldots,x_n\}$. Letting $e_k$ be the $k$-th elementary symmetric polynomial, we can notate this sum as $$e_k\left(x_1,x_2,\ldots,x_n\right).$$ In general, this will be very hard to evaluate, but at least you now have a concise way of writing it down.

Answer 2

A compact notation for these "numerators" is provided by (unsigned) Stirling numbers of the first kind, often denoted $c(n,k)$ or $\begin{bmatrix} n \\ k \end{bmatrix}$, defined to be the count of permutations of $n$ things having exactly $k$ disjoint cycles.

Let harmonic number $H_n = 1+ \frac{1}{2} + \ldots + \frac{1}{n}$. Then for $n\ge 1$:

$$ H_n = \frac{1}{n!} \begin{bmatrix} n+1 \\ 2 \end{bmatrix} $$

This is Thm. 1 in "A Stirling Encounter with Harmonic Numbers" (Benjamin et al, 2002), and see also this subsection of the Wikipedia article.

The fraction so expressed is not always in reduced form (numerator and denominator are typically not coprime). The indicated numerator can be computed by a recurrence relation, cf. (7) of Benjamin et al:

$$ \begin{bmatrix} 2 \\ 2 \end{bmatrix} = 1; \begin{bmatrix} n+1 \\ 2 \end{bmatrix} = n \begin{bmatrix} n \\ 2 \end{bmatrix} + (n-1)! $$