Paths in the Square Unit Lattice

An n-bit binary string (x(1), x(2), ... , x(n)), x(i) = +/- 1 and its partial sums s(j) = x(1) + x(2) + ... + x(j) determine paths in the unit lattice {(x, y) vertical bar x, y integers} with "steps" /\. We enumerate classes of these paths satisfying restrictions. These include well-known counts (e.g., Catalan numbers and Chung-Feller numbers) and new counts, some presented as problems with solutions.