Combinatorics of injective words for Temperley-Lieb algebras

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