Counting paths on a chessboard with a barrier

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 nn movements which stay exactly kk times, 0≤k≤n+10≤k≤n+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.