All Permutations Supersequence is coNP-complete

We prove that deciding whether a given input word contains as subsequence every possible permutation of integers {1, 2, . . . , n} is coNP-complete. The coNP-completeness holds even when given the guarantee that the input word contains as subsequences all of length n−1 sequences over the same set of integers. We also show NP-completeness of a related problem of Partially Non-crossing Perfect Matching in Bipartite Graphs, i.e. to find a perfect matching in an ordered bipartite graph where edges of the matching incident to selected vertices (even only from one side) are non-crossing.