Patterns in colored circular permutations

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Problem: The natural reconstruction is: in the setting of Gray–Lanning–Wang’s colored circular pattern packing, the pattern $\rho=1_r2_b$ is the only colored circular pattern for which there exists a finite circular $\rho$-optimal permutation whose colored-block format differs from that of $\rho$. This is supported by Remark 4.5 and the Section 6 sentence referring to “optimal permutations that have a different format.”

This conjecture is false even for a pattern with three colored blocks.

Result: Let

Its cyclic shifts are

Consider the length-$5$ colored circular permutation

First, $\pi$ is circular $\rho$-optimal. In any length-$5$ colored permutation, an occurrence of a cyclic shift of $\rho$ must use one red entry and two blue entries, with the red value strictly between the two blue values. If there are $b$ blue entries, then for each red entry $x$, the number of blue pairs straddling $x$ is $L_xU_x$, where $L_x$ and $U_x$ are the numbers of blue entries below and above $x$. For $b=2,3,4$, this gives at most $3,4,4$ possible triples respectively; the other cases give $0$. Hence

But in $\pi$, the four triples

are all occurrences of the shift $1_b3_b2_r$. Thus

Second, $\pi$ does not have the same colored-block format as $\rho$. A same-format permutation for $\rho=2_r1_b3_b$ would have form

so no blue value could lie between two red values. But in $\pi$, the red values are $2,4$, while the blue value $3$ lies between them. Therefore the red entries cannot form the single red colored block required by the format of $\rho$.

Thus a pattern other than $1_r2_b$ has an optimal circular permutation of different format. The reconstructed conjecture is disproved.

Audit: the counterexample has $n=5\ge |\rho|=3$, uses only the colors appearing in $\rho$, and applies exactly the paper’s definition $f_c(\pi,\rho)=\sum_i f(\pi,S^i\rho)$.

Citation: Definitions and conjectural context: Daniel Gray, Charles Lanning, Hua Wang, “Patterns in colored circular permutations,” Involve 12 (2019), 157–169. No prior resolution is used here.