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.

本文研究了平面内射词复合体的组合性质，该平面复合体是Temperley-Lieb代数上模块的链复合体，这是由于我们的同构稳定性而产生的。尽管是线性对象而不是离散对象，但是我们的链复合体仍显示出有趣的组合特性。我们证明了该复合物的欧拉特征是第n个Fine数。我们为它的高维同源模块给出了一个表示形式的交替求和公式，并在进一步限制接地环的情况下，根据某些标准Young tableaux分解了该模块。这三项结果（受Reiner和Webb关于内射词复数的结果启发）可以看作是对n的解释。-th个精细数字，类似于n个字母的排列数目的“平面”或“戴克路径” 。这种解释在文献中有先驱，但在这里自然是出于对同源稳定性的考虑。我们的最终结果显示了我们的复数和Jacobsthal数的边界图之间的令人惊讶的联系。