We are dealing with the problem of counting the paths joining two points of a chessboard
in the presence of a barrier. The formula for counting all the paths joining two distinct
positions on the chessboard lying always over a barrier is well known (see for example
Feller (1968) [1], Kreher and Stinson (1999) [3]). The problem is here extended to the
calculation of all the possible paths of n movements which stay exactly k times, 0 k
n C 1, over the barrier. Such a problem, motivated by the study of financial options of
Parisian type, is completely solved by virtue of five different formulas depending on the
initial and final positions and on the level of the barrier.
