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I did not read your long answer at the linked thread. However, you did mention proceeding in order of number and using a recursive relation. You can do the same here. Define $Q(x,k,m)$ to be true iff there is a valid grouping $G$ of $B∪S$ for some $S⊆H$ such that exactly $k$ pieces in $S$ have numbers at most $m$ and $x$ is the multiset of colours of runs in ...


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