This equivalence provides a bijective proof that all of these kinds of objects are counted by a single integer sequence.

[5] The closely related large Schröder numbers are equal to twice the Schröder–Hipparchus numbers, and may also be used to count several types of combinatorial objects including certain kinds of lattice paths, partitions of a rectangle into smaller rectangles by recursive slicing, and parenthesizations in which a pair of parentheses surrounding the whole sequence of elements is also allowed.

As well as the summation formula above, the Schröder–Hipparchus numbers may be defined by a recurrence relation: Stanley proves this fact using generating functions[8] while Foata and Zeilberger provide a direct combinatorial proof.

[9] Plutarch's dialogue Table Talk contains the line: This statement went unexplained until 1994, when David Hough, a graduate student at George Washington University, observed that there are 103049 ways of inserting parentheses into a sequence of ten items.

Contemporary philosopher Susanne Bobzien (2011) has argued that Chrysippus's calculation was the correct one, based on her analysis of Stoic logic.