06030398

j

06030398 Knopfmacher, Arnold Mansour, Toufik Record statistics in a random composition. Discrete Appl. Math. 160, No. 4-5, 593-603 (2012). 2012 Elsevier Science B.V. (North-Holland), Amsterdam EN composition record left-to-right maxima generating function Mellin transform asymptotic estimates

doi:10.1016/j.dam.2011.10.025

Summary: A composition$\sigma =a_{1}a_{2}\ldots a_{m}$ of $n$ is an ordered collection of positive integers whose sum is $n$. An element $a_{i}$ in $\sigma $ is a strong (weak) record if $a_{i}>a_{j} (a_{i}\ge a_{j})$ for all $j=1,2,\ldots ,i - 1$. Furthermore, the position of this record is $i$. We derive generating functions for the total number of strong (weak) records in all compositions of $n$, as well as for the sum of the positions of the records in all compositions of $n$, where the parts $a_{i}$ belong to $A=[d]:={1,2,\ldots ,d}$ or $A=N$. In particular when $A=N$, we find the asymptotic mean values for the number, and for the sum of positions of records in compositions of $n$.