We investigate the evolution in time of the position of a fixed number in the insertion tableau when the Robinson-Schensted-Knuth algorithm is applied to a sequence of random numbers. When the length of the sequence tends to infinity, a typical trajectory after scaling converges uniformly in probability to some deterministic curve.

Introduction and objective. Systematic monitoring of physical activity of young populations from different regions may be useful for making international comparisons, better understanding trends in physical activity and designing public health interventions to increase physical activity. Thus, the aim of the study was to comprehensively review studies examining the level of physical activity of children and adolescents from the Visegrád countries, and indicate the percentage of children and adolescents meeting the recommended level of physical activity. Materials and method. A systematic search of the published literature was conducted. Included were reports with original research that consisted of the proportion of children and adolescents who met physical activity recommendations or guidelines. To provide a current picture of physical activity of children and adolescents, the review was restricted to studies published between January 2015-November 2018. Results. Moderate-to-vigorous physical activity recommendations were met by between 7.5%-69.4% of children and adolescents. Vigorous physical activity recommendations were met by between 33.6%-64.5% of children and adolescents. Daily recommendations for the number of steps taken were fulfilled by between 11.6%-69.0% of children and adolescents. Differences between age categories, genders and countries were observed. Conclusions. Public health interventions to increase physical activity are required to decrease insufficient levels of physical activity in children and adolescents, especially in girls. The review also highlights the need for studies with representative random samples that use validated measurement methods.

We discuss various square-free factorizations in monoids in the context of: atomicity, ascending chain condition for principal ideals, decomposition, and a greatest common divisor property. Moreover, we obtain a full characterization of submonoids of factorial monoids in which all square-free elements of a submonoid are square-free in a monoid. We also present factorial properties implying that all atoms of a submonoid are square-free in a monoid.(1.1) A(M) ⊂ S(H).

We consider the Robinson–Schensted–Knuth algorithm applied to a random input and investigate the shape of the bumping route (in the vicinity of the y-axis) when a specified number is inserted into a large Plancherel-distributed random tableau. We show that after a projective change of the coordinate system the bumping route converges in distribution to the Poisson process.

Goulden-Rattan polynomials give the exact value of the subdominant part of the normalized characters of the symmetric groups in terms of certain quantities (C i ) which describe the macroscopic shape of the Young diagram. The Goulden-Rattan positivity conjecture states that the coefficients of these polynomials are positive rational numbers with small denominators. We prove a special case of this conjecture for the coefficient of the quadratic term C 2 2 by applying certain bijections involving maps (i.e., graphs drawn on surfaces).