Splittability and 1-amalgamability of permutation classes

A permutation class $C$ is splittable if it is contained in a merge of two of its proper subclasses, and it is 1-amalgamable if given two permutations $\sigma$ and $\tau$ in $C$, each with a marked element, we can find a permutation $\pi$ in $C$ containing both $\sigma$ and $\tau$ such that the two marked elements coincide. It was previously shown that unsplittability implies 1-amalgamability. We prove that unsplittability and 1-amalgamability are not equivalent properties of permutation classes by showing that the class $Av(1423, 1342)$ is both splittable and 1-amalgamable. Our construction is based on the concept of LR-inflations, which we introduce here and which may be of independent interest.Comment: 17 pages, 7 figure

Griddings of Permutations and Hardness of Pattern Matching

We study the complexity of the decision problem known as Permutation Pattern Matching, or PPM. The input of PPM consists of a pair of permutations ? (the "text") and ? (the "pattern"), and the goal is to decide whether ? contains ? as a subpermutation. On general inputs, PPM is known to be NP-complete by a result of Bose, Buss and Lubiw. In this paper, we focus on restricted instances of PPM where the text is assumed to avoid a fixed (small) pattern ?; this restriction is known as Av(?)-PPM. It has been previously shown that Av(?)-PPM is polynomial for any ? of size at most 3, while it is NP-hard for any ? containing a monotone subsequence of length four. In this paper, we present a new hardness reduction which allows us to show, in a uniform way, that Av(?)-PPM is hard for every ? of size at least 6, for every ? of size 5 except the symmetry class of 41352, as well as for every ? symmetric to one of the three permutations 4321, 4312 and 4231. Moreover, assuming the exponential time hypothesis, none of these hard cases of Av(?)-PPM can be solved in time 2^o(n/log n). Previously, such conditional lower bound was not known even for the unconstrained PPM problem. On the tractability side, we combine the CSP approach of Guillemot and Marx with the structural results of Huczynska and Vatter to show that for any monotone-griddable permutation class ?, PPM is polynomial when the text is restricted to a permutation from ?

Lepidoptera of North America 2. Distribution of the butterflies (Papilionoidea and Hesperioidea) of the eastern United States

Chiefly maps

Fixation of Type Locality for \u3ci\u3eLycaena acmon\u3c/i\u3e Westwood and Characterization of the Species and its Distribution

Lycaena acmon Westwood, 1852 is based on a painting and plate legend in Westwood and Hewitson’s Genera of Diurnal Lepidoptera. The specimen illustrated was located in the British Museum Natural History and is the holotype by monotypy. The accompanying plate legend gives “California” as the type locality. Because the butterfly is a member of a complex of species, now considered in the genus Plebejus (Opler and Warren, 2003), that requires much systematic study, and some confusion exists on the identity of L. acmon, it is necessary to fix a more specific type locality, to characterize the species, and to present its synonymy. In addition, a number of taxa described or cited as P. acmon are listed, but which likely represent other species