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A special sorting operation called Context Directed Swap, and denoted \textbf{cds}, performs certain types of block interchanges on permutations. When a permutation is sortable by \textbf{cds}, then \textbf{cds} sorts it using the fewest possible block interchanges of any kind. This work introduces a classification of permutations based on their number of \textbf{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-\textbf{cds}-sortability of a permutation. Focusing on the classification of permutations with maximal strategic pile, a complete characterization is given when the number of \textbf{cds}-eligible contexts is close to maximal as well as when the number of eligible contexts is minimal. A group action that preserves the number of \textbf{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 \textbf{cds}-eligible contexts. Prior work introduced a natural two-person game on permutations that are not \textbf{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., Comment: 22 pages more...

The study of sorting permutations by block interchanges has recently been stimulated by a phenomenon observed in the genome maintenance of certain ciliate species. The result was the identification of a block interchange operation that applies only under certain constraints. Interestingly, this constrained block interchange operation can be generalized naturally to simple graphs and to an operation on square matrices. This more general context provides numerous techniques applicable to the original context. In this paper we consider the more general context, and obtain an enumeration, in closed form, of all simple graphs on n vertices that are ``sortable" by the graph analogue of the constrained version of block interchanges. We also obtain asymptotic results on the proportion of graphs on n vertices that are so sortable. more...

Prior studies of the efficiency of the block interchange (swap) and the reversal sorting operations on (signed) permutations identified specialized versions of the these operations. These specialized operations are here called context directed reversal, abbreviated cdr, and context directed swap, abbreviated cds. Prior works have also characterized which (signed) permutations are sortable by cdr or by cds. It is now known that when a permutation is cds sortable in n steps, then any application of n consecutive applicable cds operations will sort it. Examples show that this is not the case for cdr. This phenomenon is the focus of this paper. It is proven that if a signed permutation is cdr sortable, then any cdr fixed point of it is cds sortable (the cds Rescue Theorem). The cds Rescue Theorem is discussed in the context of a mathematical model for ciliate micronuclear decryption. It is also proven that though for a given permutation the number of cdr operations leading to different cdr fixed points may be different from each other, the parity of these two numbers is the same (the cdr Parity Theorem). This result provides a solution to two previously formulated decision problems regarding certain combinatorial games., Comment: 18 pages, 13 figures more...

We introduce a game on graphs. By a theorem of Zermelo, each instance of the game on a finite graph is determined. While the general decision problem on which player has a winning strategy in a given instance of the game is unsolved, we solve the decision problem for a specific class of finite graphs. This result is then applied to a permutation sorting game to prove the optimality of a proportional bound under which TWO has a winning strategy., Comment: 14 pages more...

Permutation sorting, one of the fundamental steps in pre-processing data for the efficient application of other algorithms, has a long history in mathematical research literature and has numerous applications. Two special-purpose sorting operations are considered in this paper: context directed swap, abbreviated cds, and context directed reversal, abbreviated cdr. These are special cases of sorting operations that were studied in prior work on permutation sorting. Moreover, cds and cdr have been postulated to model molecular sorting events that occur in the genome maintenance program of certain species of single-celled organisms called ciliates. This paper investigates mathematical aspects of these two sorting operations. The main result of this paper is a generalization of previously discovered characterizations of cds-sortability of a permutation. The combinatorial structure underlying this generalization suggests natural combinatorial two-player games. These games are the main mathematical innovation of this paper., Comment: to appear in Discrete Mathematics, Algorithms and Applications more...

No convenient internal characterization of spaces that are productively Lindelof is known. Perhaps the best general result known is Alster's internal characterization, under the Continuum Hypothesis, of productively Lindelof spaces which have a basis of cardinality at most $\aleph_1$. It turns out that topological spaces having Alster's property are also productively weakly Lindelof. The weakly Lindelof spaces form a much larger class of spaces than the Lindelof spaces. In many instances spaces having Alster's property satisfy a seemingly stronger version of Alster's property and consequently are productively X, where X is a covering property stronger than the Lindelof property. This paper examines the question: When is it the case that a space that is productively X is also productively Y, where X and Y are covering properties related to the Lindelof property., Comment: 16 pages more...