Asymptotics of the number of \(k\)-words with an \(l\)-descent
Fix three positive integers, \(k,l,n\), with \(l\leq k\). The Schensted-Knuth correspondence gives the number of words \(w=w_1w_2\cdots w_n\) with \(1\leq w_i\leq k\), for which there are subscripts \(1\leq i_1<i_2<\cdots <i_l\leq n\) such that \(w_{i_1}>w_{i_2}>\cdots >w_{i_l}\). In this paper, the author calculates the asymptotic behaviour of this number as \(n\rightarrow \infty\). The result is achieved with the aid of asymptotic results of \textit{P. S. Cohen} and \textit{A. Regev} [SIAM J. Math. Anal. 19, No. 5, 1204-1215 (1980, Zbl 0676.60030)].
scientific article; zbMATH DE number 1135582