Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k

In the setting of The Real Numbers: Standing Notation and Background, let $\mathbb{N}$ be the natural numbers, carried into $\mathbb{R}$ by the natural-number image, and for $k\in\mathbb{N}$ let $[k]$ be the initial segment determined by $k$. Let $d\in\mathbb{N}$. For $k\in\mathbb{N}$ let $[2d]^{k}$ be the set of $k$-tuples in $[2d]$, which are the words of length $k$ in the letters $1,\dots,2d$, let $W^{\circ}_{d,k}\subseteq[2d]^{k}$ be the set of cyclically reduced words of length $k$, and let $(2d)^{k}$ denote the natural power in $\mathbb{R}$. Finite sets and sums over them are those of Finite Set and Sum over a Finite Index Set.

Then the following hold for every $k\in\mathbb{N}$.

1. (Finiteness)¶ The sets $[2d]^{k}$ and $W^{\circ}_{d,k}$ are nonempty and finite, and so is every nonempty subset of $[2d]^{k}$.

2. (Counting bound)¶ Let $B\subseteq[2d]^{k}$ be nonempty, let $M\in\mathbb{R}$, and let $a:B\to\mathbb{R}$ satisfy $0\le a(w)\le M$ for every $w\in B$. Then

$$\sum_{w\in B}a(w)\le M\,(2d)^{k}.$$