The number of minimal lattice paths restricted by two parallel lines

Abstract

We deal with non-decreasing paths on the non-negative quadrant of the integral square lattice, called by minimal lattice paths, from (0,0) to a point ( n, m) restricted by two parallel lines with an incline k (⩾0). We express the generating functions of the number of these distinct minimal lattice paths in terms of the polynomials ϕ k(n, x) = ∑ l = 0 [ n k ] n − kl l (−x) l, n⩾) Formulas obtained thus include the generating function of the so-called higher Catalan number C k ( n) or Ballot numbers as the special case. The number of minimal lattice paths for k=1 is given as an explicit form by expanding the corresponding generating function.