Combinatorics of Injective words for temperley-lieb algebras

Rachel Boyd* (Corresponding Author), Richard Hepworth

*Corresponding author for this work

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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.
Original languageEnglish
Article number105446
Number of pages27
JournalJournal of Combinatorial Theory, Series A
Early online date18 Mar 2021
Publication statusPublished - Jul 2021


  • Temperley-Lieb Algebras
  • Fine numbers
  • Jacobsthal numbers
  • Chain complexes
  • Temperley-Lieb algebras


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