  We initiate the study of model structures on (categories induced by) lattice
posets, a subject we dub homotopical combinatorics. In the case of a finite
total order $[n]$, we enumerate all model structures, exhibiting a rich
combinatorial structure encoded by Shapiro's Catalan triangle. This is an
application of previous work of the authors on the theory of $N_\infty$-operads
for cyclic groups of prime power order, along with new structural insights
concerning extending choices of certain model structures on subcategories of
$[n]$.
