Combinatorics of injective words for Temperley-Lieb algebras

TL;DR

This paper explores the combinatorial structure of a chain complex over the Temperley-Lieb algebra, revealing connections to Fine numbers, Young tableaux, and Jacobsthal numbers, with implications for homological stability and combinatorics.

Contribution

It introduces a novel combinatorial analysis of the complex of planar injective words, linking it to Fine numbers, Young tableaux, and Jacobsthal numbers, and extends understanding of homological stability.

Findings

Euler characteristic equals the n-th Fine number

Decomposition of top homology in terms of Young tableaux

Boundary maps relate to Jacobsthal numbers

Abstract

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.