Problems related to type-A and type-B matrices of chromatic joins

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Problem: Reconstructed statement: for $n\ge 1$, let $M_n=NC_2(2n)$ be the set of noncrossing perfect matchings of $\{1,\dots,2n\}$, with vertices in cyclic order. For $\alpha,\beta\in M_n$, let

denote the number of connected components of the graph on $\{1,\dots,2n\}$ whose edges are the pairs of $\alpha$ and $\beta$, equivalently the number of blocks of the join $\alpha\vee\beta$. The problem asks for a fixed-point-free permutation $\sigma:M_n\to M_n$ such that

As stated for $n=1$, this is false: $M_1$ has one element, so no derangement exists. The minimal natural repair is $n\ge 2$.

Result: For every $n\ge 2$, the cyclic rotation gives the desired derangement.

Let $\rho(i)=i+1\pmod {2n}$, with $2n+1=1$, and define

Rotation preserves noncrossing matchings, so $\sigma$ is a permutation of $M_n$.

We prove $\langle \alpha,\rho(\alpha)\rangle=1$. Suppose the union graph of $\alpha$ and $\rho(\alpha)$ is disconnected, and color one connected component black and all other vertices white. Then for every edge $\{i,j\}\in\alpha$,

and since $\{\rho(i),\rho(j)\}\in\rho(\alpha)$,

Claim: if $\alpha\in NC_2(2n)$ and a coloring satisfies both equalities for every $\{i,j\}\in\alpha$, then all colors are equal.

Proof by induction on $n$. Every noncrossing perfect matching has an adjacent pair. After cyclic relabeling, assume $\{1,2\}\in\alpha$. Then

so $c_1=c_2=c_3$. Delete vertices $1,2$ and relabel $3,\dots,2n$ as $1,\dots,2n-2$. The remaining matching is again noncrossing, and the induced coloring still satisfies the same two equality conditions; the only wraparound case uses $c_1=c_3$. By induction all remaining colors are equal, hence all original colors are equal. This proves the claim.

Thus the supposed component coloring must be constant, so the union graph is connected. Hence

Finally, for $n\ge 2$, $\rho(\alpha)\ne\alpha$. Indeed, if $\rho(\alpha)=\alpha$, then

impossible for $n\ge 2$. Therefore $\sigma=\rho$ is a derangement with the required property.

Verification audit: the only repair is the necessary exclusion of the singleton case $n=1$. The construction uses no extra hypotheses beyond noncrossing perfect matchings in cyclic order, and the conclusion is exactly $\langle\alpha,\sigma(\alpha)\rangle=1$ for every $\alpha\in M_n$.

Citation: No external resolution is invoked; the construction and proof above are elementary.