Limit Behavior of Maxima in Geometric Words representing Set Partitions

We consider geometric words ω1 · · ·ωn with letters satisfying the restricted growth property ωk ≤ d + max{ω0, . . . , ωk−1}, where ω0 := 0 and d ≥ 1. For d = 1 these words are in 1-to-1 correspondence with set partitions and for this case we show that the number of left-to-right maxima (suitable centered) does not converge to a fixed limit law as n tends to infinity. This becomes wrong for d ≥ 2 for which we prove that convergence does occur and the limit law is normal. Moreover, we also consider related quantities such as the value of the maximal letter and the number of maximal letters and show again non-convergence to a fixed limit law.