# Sort Like a Ciliate: Solutions and Strategies in a Genomic Game

## Additional Funding Sources

This research, conducted at the Complexity Across Disciplines Research Experience for Undergraduates site, was supported by the National Science Foundation under Grant No. DMS-1659872 and by Boise State University.

## Abstract

In 2003, Prescott et al. hypothesized a special sorting operation active during ciliate genome maintenance. This operation, called cds, involves block interchanges of “permutations”, or unsorted lists. Christie (1996) discovered that for permutations sortable by cds, cds sorts them using the fewest possible block interchanges of any kind. Adamyk et al. (2013) discovered an efficient way of quantifying the non-cds-sortability of a permutation called the strategic pile. We investigate permutations with maximal strategic pile, aiming to determine when such a permutation has a given number of available cds moves. We complete this characterization when the number of available moves is close to maximal and when the number of available moves is minimal.

We discover a collection of symmetries on these permutations that preserves the number of available cds moves and the maximality of the strategic pile. We then study permutations that are “symmetric”, count the number of permutations with maximal strategic pile and a given number of symmetries, and rediscover a classical theorem of Wilson about prime numbers.

Adamyk et al. (2013) discovered a natural two player game using permutations that are not cds-sortable. We discover new sufficient conditions for player ONE to have a winning strategy in this game.

Sort Like a Ciliate: Solutions and Strategies in a Genomic Game

In 2003, Prescott et al. hypothesized a special sorting operation active during ciliate genome maintenance. This operation, called cds, involves block interchanges of “permutations”, or unsorted lists. Christie (1996) discovered that for permutations sortable by cds, cds sorts them using the fewest possible block interchanges of any kind. Adamyk et al. (2013) discovered an efficient way of quantifying the non-cds-sortability of a permutation called the strategic pile. We investigate permutations with maximal strategic pile, aiming to determine when such a permutation has a given number of available cds moves. We complete this characterization when the number of available moves is close to maximal and when the number of available moves is minimal.

We discover a collection of symmetries on these permutations that preserves the number of available cds moves and the maximality of the strategic pile. We then study permutations that are “symmetric”, count the number of permutations with maximal strategic pile and a given number of symmetries, and rediscover a classical theorem of Wilson about prime numbers.

Adamyk et al. (2013) discovered a natural two player game using permutations that are not cds-sortable. We discover new sufficient conditions for player ONE to have a winning strategy in this game.

## Comments

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