Generalizing the combinatorics of binomial coefficients via l-nomials

An l-sequence is defined by an = lan−1 − an−2, with initial conditions a0 = 0, a1 = 1. These l-sequences play a remarkable role in partition theory, allowing l-generalizations of the Lecture Hall Theorem [7, 8] and Euler’s Partition Theorem [8, 26]. These special properties are not shared with other sequences, such as the Fibonacci sequence, defined by second-order linear recurrences. The l-sequence gives rise to the l-nomial coefficient `