Consider all possible arrangements of the sequence $S_k=(\underbrace{1,1,...,1}_{k},\underbrace{2,2,...,2}_{k}),$ where $k\ge 1.$ There's a neat proof, using indicators and linearity of expectation, that the number of maximal runs in an arrangement of $S_k$ chosen uniformly at random has expectation $k+1.$
However, if we let $A(k,r)$ denote the number of arrangements of $S_k$ that contain exactly $r$ maximal runs ($2\le r\le 2k$), then the same result is an immediate consequence of $A(k,\cdot)$ being symmetrical about $r=k+1$ for each $k\ge 1.$ So, although there's already a nice proof of the expectation, I'm interested in a simple-as-possible alternative proof by way of proving this symmetry.
Now, the symmetry can be established in a round-about way by noting an obvious one-to-one correspondence between $r$-run arrangements of $S_k$ on the one hand, and on the other hand, $(r-1)$-turn lattice-paths between $(0,0)$ and $(k,k)$ consisting only of NE/EN turns. A paper by Ting Kuo$^\dagger$ derives a general result for the latter, a special case of which gives the following explicit formula for $A(k,r)$: $$A(k,r)=\begin{cases} 2\dbinom{k-1}{r-3\over 2}\dbinom{k-1}{r-1\over 2}&\text{if $r$ is odd}\\[2ex] 2\dbinom{k-1}{{r\over 2}-1}^2&\text{if $r$ is even} \end{cases}\quad\quad\text{where }\quad k\ge 1,\quad 2\le r\le 2k.$$
(This is equivalent to a formula given without references at OEIS A152659.) From the formula it is straightforward, using well-known properties of binomial coefficients, to show that indeed $A(k,r)=A(k,2k-(r-2))$. That is, the required symmetry holds; hence, the expectation of the number of runs is the middle value in $\{2,...,2k\}$, which is $k+1.$
That formula is great, but to me it seems a vast overkill for merely establishing the symmetry of $A(k,\cdot)$; hence, my question:
How might the symmetry of $A(k,\cdot)$ be more-directly established than by recourse to its explicit formula?
$^\dagger$Ting Kuo, Enumeration of 2D Lattice Paths with a Given Number of Turns, J. of Computers, Vol. 26, No. 4, January 2016.