  This paper studies combinatorial properties of the 'complex of planar
injective words', a chain complex of modules over the Temperley-Lieb algebra
that arose in our work on homological stability. Despite being a linear rather
than a discrete object, our chain complex nevertheless exhibits interesting
combinatorial properties. We show that the Euler characteristic of this complex
is the n-th Fine number. We obtain an alternating sum formula for the
representation given by its top-dimensional homology module and, under further
restrictions on the ground ring, we decompose this module in terms of certain
standard Young tableaux. This trio of results - inspired by results of Reiner
and Webb for the complex of injective words - can be viewed as an
interpretation of the n-th Fine number as the 'planar' or 'Dyck path' analogue
of the number of derangements of n letters. This interpretation has precursors
in the literature, but here emerges naturally from considerations in
homological stability. Our final result shows a surprising connection between
the boundary maps of our complex and the Jacobsthal numbers.
