Level number sequences of trees and the lambda algebra (Q1178028)

Generalizing results of \textit{P. Flajolet} and \textit{H. Prodinger} [Discrete Math. 65, 149-165 (1987; Zbl 0634.05021)], the author solves in terms of generating function the following enumeration problem: What is the number of sequences of positive integers of length \(\ell\), sum \(n\), final integer \(j\), initial integer \(\leq m\), satisfying the inequality \(i_{k+1}\leq di_ k\), for each \(1\leq k\leq\ell-1\)? Several asymptotic results relevant for the lambda algebra, see \textit{M. C. Tangora} [Computing the homology for the lambda algebra, Mem. Am. Math. Soc. 337 (1985; Zbl 0584.55019)] are derived from the generating function. Another application is counting level number sequences of \(d\)-ary trees.