Up-smooth samples of geometric variables

We study samples T = (T₁.....T_n) of length n where the letters F_i are independently generated according to the geometric distribution F(F_j = i) = pq^i-1, for 1≤ j ≤n, with p + q=1 and 0<p<1. An up-smooth sample F is a sample such that F _i+1-F_i ≤1. We find generating functions for the probability that a sample of n geometric variables is up-smooth, with or without a specified first letter. We also extend the up-smooth results to words over an alphabet of k letters and to compositions of integers. In addition we study smooth samples F of geometric random variables, where the condition now is F_i+1F<_i |≤ 1.