There are many questions and answers throughout concerning longest run of heads after $n$ flips of a fair coin. For example, this question has multiple detailed answers. A good reference listed was "The Longest Run of Heads" by Schilling where:
Consider $n$ independent tosses of a fair coin, and let $R_n$ represent the length of the longest run of heads. [...] $$E[R_n] \sim \log_2[n]$$
My Question: What if the coins are not arranged in a linear fashion, but in a circle? For example, HHTH has longest length of heads as 2 (linear), but arranged in a circle you can also consider it to be HHHT which has longest length of 3. Alternatively, can also think of two-colored beads on a necklace.
For numerical experiments:
Longest Run, with counts of when $R_n=k$ for $k=\{0,\dots,20\}$ among all $2^{20}$ combinations of flipping 20 coins
{1, 17710, 205606, 324020, 239231, 133751, 67249, 32392, 15309, 7163, \
3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}
And expected value of 3.72925.
For second experiment, for each of the $2^{20}$ combinations looked at $R_n$ for all 20 possible 'rotations' of the circles of coins. If you define $S_n=\operatorname{max}(R_n)$ to be taken over all rotations, here are the counts when $S_n=k$ for $k=\{0,\dots,20\}$
{1, 15126, 181204, 305049, 242395, 145440, 78240, 40280, 20370, \
10230, 5120, 2560, 1280, 640, 320, 160, 80, 40, 20, 20, 1}
And expected value of 3.9182
Example: 145,440 combinations out of $2^{20}$ had longest length of heads be 5 when all 20 coins where in a circle, or two-colored beads in a necklace.
Is $S_n \sim \log_2[n]$ also as well ? I found many references on longest run of heads, but did not see any when arranged in a circle were also taken into account.