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A sock ordering is a sequence of socks with different colors. A sock orderingis foot-sortable if the sequence of socks can be sorted by a stack so thatsocks with the same color form a contiguous block. The problem of decidingwhether a given sock ordering is foot-sortable was first considered by Defantand Kravitz, who resolved the case for alignment-free 2-uniform sock orderings.In this paper, we resolve the problem in a more general setting, where eachcolor appears in the sock ordering at most twice. A key component of theargument is a fast algorithm that determines the foot-sortability of a sockordering of length $N$ in time $O(N\log N)$, which is also an interestingresult on its own.

Deciding Foot-sortability and Minimal 2-bounded Non-foot-sortable Sock Orderings