Drew Armstrong ; Brendon Rhoades ; Nathan Williams - Rational Catalan Combinatorics: The Associahedron

dmtcs:2355 - Discrete Mathematics & Theoretical Computer Science, January 1, 2013, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) - https://doi.org/10.46298/dmtcs.2355
Rational Catalan Combinatorics: The AssociahedronArticle

Authors: Drew Armstrong 1; Brendon Rhoades 2; Nathan Williams ORCID3

  • 1 Department of Mathematics [Miami]
  • 2 Department of Mathematics [Univ California San Diego]
  • 3 Department of Mathematics [Minneapolis]

Each positive rational number $x>0$ can be written $\textbf{uniquely}$ as $x=a/(b-a)$ for coprime positive integers 0<$a$<$b$. We will identify $x$ with the pair $(a,b)$. In this extended abstract we use $\textit{rational Dyck paths}$ to define for each positive rational $x>0$ a simplicial complex $\mathsf{Ass} (x)=\mathsf{Ass} (a,b)$ called the $\textit{rational associahedron}$. It is a pure simplicial complex of dimension $a-2$, and its maximal faces are counted by the $\textit{rational Catalan number}$ $\mathsf{Cat} (x)=\mathsf{Cat}(a,b):=\frac{(a+b-1)! }{ a! b!}.$ The cases $(a,b)=(n,n+1)$ and $(a,b)=(n,kn+1)$ recover the classical associahedron and its Fuss-Catalan generalization studied by Athanasiadis-Tzanaki and Fomin-Reading. We prove that $\mathsf{Ass} (a,b)$ is shellable and give nice product formulas for its $h$-vector (the $\textit{rational Narayana numbers}$) and $f$-vector (the $\textit{rational Kirkman numbers}$). We define $\mathsf{Ass} (a,b)$ .


Volume: DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)
Section: Proceedings
Published on: January 1, 2013
Imported on: November 21, 2016
Keywords: associahedron,Dyck path,$f$-vector,$h$-vector,shelling,noncrossing partition,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • Combinatorics of Root Systems; Funder: National Science Foundation; Code: 1001825
  • Combinatorics and Representation Theory; Funder: National Science Foundation; Code: 1068861

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