Mathematically, Gilbreath shuffles can be described by Gilbreath permutations, permutations of the numbers from 1 to n that can be obtained by a Gilbreath shuffle with a deck of cards labeled with these numbers in order.
A Gilbreath shuffle may be uniquely determined by specifying which of the positions in the resulting shuffled deck are occupied by cards that were dealt off into the second pile, and which positions are occupied by cards that were not dealt off.