Consider the free simple random walk on \(\mathbb{Z}^ 2\), i.e. its steps are \((1,0)\), \((0,1)\), \((-1,0)\) or \((0,-1)\). Starting from results of E. Csáki, S. G. Mohanty and the first author [ibid. 29, 309-318 (1990; Zbl 0701.60070)], the authors compute numbers of conditioned paths of length \(d\) from \((0,0)\) to \((a,b)\), where \((a,b)\) has several different (fixed) positions with respect to the line \(L = \{(x,y) : y = x + k\}\).
Examples: The number of these paths entering \(L\) exactly \(r\) times; crossing it exactly \(r\) times; never crossing it and entering it \(r\) times from below; entering it \(r\) times from above. Similar numbers where the \(i\)th arrival in \(L\) occurs after \(d_ 1\) steps. Numbers of paths with prescribed numbers of arrivals and crossings; also with prescribed number of steps above \(L\).