Adjacent Divisibility in the Narayana Triangle

This paper studies divisibility between neighboring entries in rows of the Narayana triangle. The quotient of two adjacent Narayana numbers is a quotient of triangular numbers, reducing the problem to divisibility between triangular numbers. We give an exact divisibility criterion and determine the natural density in every fixed column. For the row statistic a(n) counting nontrivial adjacent divisibilities in the increasing half of row n, we prove a first-order asymptotic formula for its summatory function. We also show that arbitrarily long chains of consecutive divisibilities occur in infinitely many rows. Finally, fixed adjacent quotients lead to generalized Pell equations, with a particularly simple characterization for quotient 2.