SHIFT TRANSFORM APPROACH TO THE TWO-SIDED BALLOT THEOREM

Abstract

We present a recursive formula for the two-sided ballot theorem using left and right shift transforms. In particular, we showed that the xth entry of the image of the d + 1 dimensional unit vector under the sum of the left and right shift operators is the number of walks in the lattice interval [0,d] that start at the origin and stop at the location x. This approach enables us to write a recursive formula for the number of possible n−walks between two obstacles that stop at a predetermined location.