## Abstract

Let \(M = (s_1, s_2, \ldots , s_n) \) be a sequence of distinct symbols and \(\sigma \) a permutation of \(\{1,2, \ldots , n\}\). Denote by \(\sigma (M)\) the permuted sequence \((s_{\sigma (1)}, s_{\sigma (2)}, \ldots , s_{\sigma (n)})\). For a given positive integer *d*, we will say that \(\sigma \) is *d**-resilient* if no matter how *d* entries of *M* are removed from *M* to form \(M'\) and *d* entries of \(\sigma (M)\) are removed from \(\sigma (M)\) to form \(\sigma (M)'\) (with no symbol being removed from both sequences), it is always possible to reconstruct the original sequence *M* from \(M'\) and \(\sigma (M)'\). Necessary and sufficient conditions for a permutation to be *d*-resilient are established in terms of whether certain auxiliary graphs are acyclic. We show that for *d*-resilient permutations for [*n*] to exist, *n* must have size at least exponential in *d*, and we give an algorithm to construct such permutations in this case. We show that for each *d* and all sufficiently large *n*, the fraction of all permutations on *n* elements which are *d*-resilient is bounded away from 0.

## Keywords

Deletion channel Recovery Permutations Double path graph## Mathematics Subject Classification

94B25 05C38## Notes

### Acknowledgements

Utkrisht Rajkumar thanks Young-Han Kim for support and guidance. Noga Alon and Ron Graham thank the Simons Institute for the Theory of Computing at UC Berkeley, where part of this work was done. The authors thank the referees for feedback on an earlier version of this paper.

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