We establish an improved lower bound of 10.271 for the exponential growth rate of the class of permutations avoiding the pattern 1324, and an improved upper bound of 13.5. These results depend on a new exact structural characterisation of 1324-avoiders as a subclass of an infinite staircase grid class, together with precise asymptotics of a small domino subclass whose enumeration we relate to W...

We establish an improved lower bound of 10.271 for the exponential growth rate of the class of permutations avoiding the pattern 1324, and an improved upper bound of 13.5. These results depend on a new exact structural characterisation of 1324-avoiders as a subclass of an infinite staircase grid class, together with precise asymptotics of a small “domino” subclass whose enumeration is related t...

We consider the enumeration of pattern-avoiding involutions, focusing in particular on sets defined by avoiding a single pattern of length 4. We directly enumerate the involutions avoiding 1342 and the involutions avoiding 2341. As we demonstrate, the numerical data for these problems exhibits some surprising behavior. This strange behavior even provides some very unexpected data related to the...

Cryotherapy has recently been examined as a potential treatment for Alopecia Areata (AA). AA can be managed with intralesional or systemic steroids but relapse rates among those longstanding disease are high. This review serves to describe the existing studies evaluating cryotherapy of and examine comparing steroid injection AA. A systematic literature from 1990 2022 was conducted looking keywo...

A new Sc3(C2)(CN)@Ih-C80 metallofullerene encaging a scandium carbide/cyanide alloyed cluster was prepared and investigated. Sc3(C2)(CN)@Ih-C80 was synthesized by the arc-discharging method and isolated by HPLC. Its experimental (13)C NMR spectrum with two signals clearly confirms an icosahedral C80 cage, and theoretically calculated (13)C NMR peaks agree well with the experimental results. Fur...

We prove that the number of 1324-avoiding permutations of length n is less than (7 + 4 √ 3). The novelty of our method is that we injectively encode such permutations by a pair of words of length n over a finite alphabet that avoid a given factor.

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.