A bijection between the triangulation of the associahedron and the parking functions

In the late 1950s, Sugawara successfully tackled the problem of characterizing spaces homotopically equivalent to a space of loops. He established a recognition principle based on homotopic invariants. Later, in 1961, J. Stasheff introduced the associahedron as a tool to simplify the Sugawara recognition principle. In 2007, J.-L. Loday described a triangulation of the associahedron of dimension n using \( (n + 1)^{n-1} \) simplices. Loday pointed out that this triangulation is in bijection with the number of parking functions. A parking function is a sequence \( (a_1, \ldots, a_n) \) of integers that, when sorted as \( (b_1 \leq \ldots \leq b_n) \), satisfies \( b_i \leq i \) for all \( 1 \leq i \leq n \). Loday did not provide a bijective function between the two sets. In this work, we describe in detail the bijection between the triangulation and the parking functions proposed by J.-L. Loday, based on the work of P. Rosero, which constructed a bijection between maximal chains of uncrossed partitions with nested partition chains that induce the desired triangulation. To achieve this, we think on triangulating the permutahedron \( P^{n} \) by leveraging its Coxeter system structure, and the triangulation of the associahedron is obtained by collapsing specific simplices in the triangulation of \( P^{n} \). We aim to present a more precise formulation of the method developed by P. Rosero.

• المؤلف الرئيسي: Zhigue Álvarez, Dayana Mishel (author)
• التنسيق: bachelorThesis
• اللغة: eng
• منشور في: 2024
• الموضوعات: Permutahedron; Particiones no cruzadas; Funciones de estacionamiento; Non-croosing partitions; Parking functions
• الوصول للمادة أونلاين: http://repositorio.yachaytech.edu.ec/handle/123456789/761