Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations

Defant, Engen, and Miller defined a permutation to be uniquely sorted if it has exactly one preimage under West's stack-sorting map. We enumerate classes of uniquely sorted permutations that avoid a pattern of length three and a pattern of length four by establishing bijections between these classes and various lattice paths. This allows us to prove nine conjectures of Defant.Comment: 18 pages, 16 figures, new version with updated abstract and reference

Structure of Demographic Types of Small Towns in Poland Spatial and Temporal Approach

The aim of this paper is to determine changes in the structure of demographic types of small towns in Poland between 2004 and 2013. It is assumed in the paper, following the Central Statistical Office of Poland, that small towns are urban settlements having less than 20,000 inhabitants. The time period covered in this study is the time of Poland’s accession to the EU, which brought reduction of many barriers on the labour market and in migration movement. Demographic types of small towns were determined using Webb’s typology. Natural increase and migration indicators constitute its base. It was found that the share of towns of progressive character decreased and the share of those of regressive character increased in the analyzed period. A negative migration balance had the greatest effect on the number of inhabitants of the analyzed settlement units. The described demographic changes in small towns in Poland were connected with the second stage of demographic transition

TORQUES AND SYNCHRONIZATION OF MUSCLE WORK IN LEG IN THE MARATHON RUN

NI