We study a class of observables in four-dimensional superconformal Yang--Mills theories which, in the planar limit at finite 't Hooft coupling, can be expressed as determinants of semi-infinite matrices built from Bessel functions. This determinant representation points to an underlying integrable structure, which we make explicit by showing that the observables satisfy a nonlinear differential-difference equation. We argue that the solution to this equation admits an expansion in terms of iterated Chen integrals of uniform transcendental weight. Remarkably, the coefficients in this expansion are universal positive integers, independent of the particular observable, suggesting a hidden combinatorial origin. Building on this observation, we show that the resulting expressions possess a natural interpretation in enumerative combinatorics: they coincide with the partition function (or generating function) of an ensemble of lattice paths constrained to a nontrivial domain. This correspondence extends and generalizes the classical Dyck paths to a richer family of path ensembles relevant in gauge theory.