arXiv:2011.00706v1 [math.CO] 2 Nov 2020 CLASSIFYING PERMUTATIONS UNDER CONTEXT-DIRECTED SWAPS AND THE CDS GAME G. BROWN, A. MITCHELL, R. RAGHAVAN, J. ROGGE, AND M. SCHEEPERS Abstract. A special sorting operation called Context Directed Swap, and denoted cds, performs certain types of block interchanges on permutations. When a permutation is sortable by cds, then cds sorts it using the fewest possible block interchanges of any kind. This work introduces a clas- sification of permutations based on their number of cds-eligible contexts. In prior work an object called the strategic pile of a permutation was discovered and shown to provide an efficient measure of the non-cds-sortability of a permutation. Focusing on the classification of permutations with maximal strategic pile, a complete characterization is given when the number of cds-eligible con- texts is close to maximal as well as when the number of eligible contexts is minimal. A group action that preserves the number of cds-eligible contexts of a permutation provides, via the orbit-stabilizer theorem, enumerative results regarding the number of permutations with maximal strategic pile and a given number of cds-eligible contexts. Prior work introduced a natural two-person game on per- mutations that are not cds-sortable. The decision problem of which player has a winning strategy in a particular instance of the game appears to be of high computational complexity. Extending prior results, this work presents new conditions for player ONE to have a winning strategy in this combinatorial game. 1. Introduction The sortability of permutations, non-repetitve arrays of integers, is of interest to a variety of fields including scientific computing and genetics. We study a a particular block-interchange sorting operation postulated to occur in the genomic sorting of single-celled organisms called ciliates [10]. This operation participates in decrypting the ciliate’s micronuclear genome to construct a new macronucleus. We refer to this block-interchange operation as cds, abbreviating “context directed swap”. A permutation π is cds-sortable if successive applications of cds on π results in the identity permutation. Not all permutations are cds-sortable. The criteria for sortability were studied previously in [1, 8, 9] and others. In [3] Christie discovered that cds is a minimal block-interchange, sorting a cds-sortable permutation using the fewest possible block interchanges. If a permutation is not cds-sortable, successive applications of cds will result in a permutation where cds no longer applies. Define a permutation where cds does not apply as a cds fixed point. The set of cds fixed points reachable from a permutation π, excluding the identity, is called the strategic pile of the permutation, and is denoted SP (π). An interesting phenomenon arises when a permutation is not cds-sortable; the fixed point of a permutation reached after applications of cds may depend on the order in which the cds operations were applied. This phenomenon on non cds-sortable permutations gives rise to a combinatorial game previously investigated in [1] and [7]. In this game player ONE is assigned a subset of the strategic pile. The symbol CDS(π,A) denotes this game on permutation π where A is the subset of the strategic pile assigned to player ONE. Beginning with ONE, players ONE and TWO take turns successively applying cds to π until a fixed point is reached. If the fixed point is in A, ONE wins; otherwise, TWO wins. Note that the number of moves in the game CDS(π,A) is bounded by the length of π Date : October 31, 2020. 2010 Mathematics Subject Classification. 05A05, 05A05, 91A46, 68P10. Key words and phrases. permutation sorting, group theory, combinatorics, game theory. 1