  $\lambda\upsilon$ is an extension of the $\lambda$-calculus which
internalises the calculus of substitutions. In the current paper, we
investigate the combinatorial properties of $\lambda\upsilon$ focusing on the
quantitative aspects of substitution resolution. We exhibit an unexpected
correspondence between the counting sequence for $\lambda\upsilon$-terms and
famous Catalan numbers. As a by-product, we establish effective sampling
schemes for random $\lambda\upsilon$-terms. We show that typical
$\lambda\upsilon$-terms represent, in a strong sense, non-strict computations
in the classic $\lambda$-calculus. Moreover, typically almost all substitutions
are in fact suspended, i.e. unevaluated, under closures. Consequently, we argue
that $\lambda\upsilon$ is an intrinsically non-strict calculus of explicit
substitutions. Finally, we investigate the distribution of various redexes
governing the substitution resolution in $\lambda\upsilon$ and investigate the
quantitative contribution of various substitution primitives.
