arXiv subjects
Publications and source records attributed to M. D. Atkinson.
A deflatable permutation class is one in which the simple permutations are contained in a proper subclass. Deflatable permutation classes are often easier to describe and enumerate than non-deflatable ones. Some theorems which guarantee non-deflatability are proved and examples of both deflatable and non-deflatable principal classes are given.
Isomorphisms p between pattern classes A and B are considered. It is shown that, if p is not a symmetry of the entire set of permutations, then, to within symmetry, A is a subset of one a small set of pattern classes whose structure, including their enumeration, is determined.
We enumerate three specific permutation classes defined by two forbidden patterns of length four. The techniques involve inflations of geometric grid classes.
The structure of three pattern classes Av(2143, 4321), Av(2143, 4312) and Av(1324, 4312) is determined using the machinery of monotone grid classes. This allows the permutations in these classes to be described in terms of simple diagrams and regular languages and, using this, the rational generating functions which enumerate these classes are determined.
When a set of permutations comprising a pattern class C is submitted as input to a priority queue the resulting output is again a pattern class C'. The basis of C' is determined for pattern classes C whose basis elements have length 3, and is finite in these cases. An example is given of a class C with basis 2431 for which C is not finitely based.
A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope \pm1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions. Furthermore, we show that these properties are inherited by the subclasses (under permutation involvement) of such classes, and establish the basic lattice theoretic properties of the collection of all such subclasses.
Let B be the operation of re-ordering a sequence by one pass of bubble sort. We completely answer the question of when the inverse image of a principal pattern class under B is a pattern class.
We enumerate the pattern class Av(2143,4231) and completely describe its permutations. The main tools are simple permutations and monotone grid classes.
A poset is {\it $(\3+\1)$-free} if it contains no induced subposet isomorphic to the disjoint union of a 3-element chain and a 1-element chain. These posets are of interest because of their connection with interval orders and their appearance in the $(\3+\1)$-free Conjecture of Stanley and Stembridge. The dimension 2 posets $P$ are exactly the ones which have an associated permutation $π$ where $i\prec j$ in $P$ if and only if $i<j$ as integers and $i$ comes before $j$ in the one-line notation of $π$. So we say that a permutation $π$ is {\it $(\3+\1)$-free} or {\it $(\3+\1)$-avoiding} if its poset is $(\3+\1)$-free. This is equivalent to $π$ avoiding the permutations 2341 and 4123 in the language of pattern avoidance. We give a complete structural characterization of such permutations. This permits us to find their generating function.
We prove that all subclasses of the separable permutations not containing Av(231) or a symmetry of this class have rational generating functions. Our principal tools are partial well-order, atomicity, and the theory of strongly rational permutation classes introduced here for the first time.
Pattern classes which avoid 321 and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns. The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth rates.
The concept of descent algebras over a field of characteristic zero is extended to define descent algebras over a field of prime characteristic. Some basic algebraic structure of the latter, including its radical and irreducible modules, is then determined. The decomposition matrix of the descent algebras of Coxeter group types $A$, $B$, and $D$ are calculated, and used to derive a description of the decomposition matrix of an arbitrary descent algebra. The Cartan matrix of a variety of descent algebras over a finite field is then obtained.
The descent algebra of the symmetric group, over a field of non-zero characteristic p, is studied. A homomorphism into the algebra of generalised p-modular characters of the symmetric group is defined. This is then used to determine the radical, and its nilpotency index. It also allows the irreducible representations of the descent algebra to be described.
We outline the need for stricter requirements for unconditionally secure cryptographic protocols inspired by the Russian Cards problem. A new requirement CA4 is proposed that checks for bias in single card occurrence in announcements consisting of alternatives for players' holdings of cards. This requirement CA4 is shown to be equivalent to an alternative requirement CA5. All announcements found to satisfy CA4 are 2-designs. We also show that all binary designs are 3-designs. Instead of avoiding bias in announcements produced by such protocols, one may as well apply unbiased protocols such that patterns in announcements become meaningless. We gave two examples of such protocols for card deal parameters (3,3,1), i.e. two of the players hold three cards, and the remaining player, playing the role of eavesdropper, holds a single card.
A pattern class is a set of permutations closed under the formation of subpermutations. Such classes can be characterised as those permutations not involving a particular set of forbidden permutations. A simple collection of necessary and sufficient conditions on sets of forbidden permutations which ensure that the associated pattern class is of polynomial growth is determined. A catalogue of all such sets of forbidden permutations having three or fewer elements is provided together with bounds on the degrees of the associated enumerating polynomials.
Pattern avoidance classes of permutations that cannot be expressed as unions of proper subclasses can be described as the set of subpermutations of a single bijection. In the case that this bijection is a permutation of the natural numbers a structure theorem is given. The structure theorem shows that the class is almost closed under direct sums or has a rational generating function.
A simple permutation is one which maps no proper non-singleton interval onto an interval. We consider the enumeration of simple permutations from several aspects. Our results include a straightforward relationship between the ordinary generating function for simple permutations and that for all permutations, that the coefficients of this series are not P-recursive, an asymptotic expansion for these coefficients, and a number of congruence results.
A connection between permutations that avoid 4231 and a certain queueing discipline is established. It is proved that a more restrictive queueing discipline corresponds to avoiding both 4231 and 42513, and enumeration results for such permutations are given.