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Rational associahedra and noncrossing partitions | | Drew Armstrong
; Brendon Rhoades
; Nathan Williams
; | | Date: |
31 May 2013 | | Abstract: | Each positive rational number x>0 can be written uniquely as x=a/(b-a) for
coprime positive integers 0<a<b. We will identify x with the pair (a,b). In
this paper we define for each positive rational x>0 a simplicial complex
Ass(x)=Ass(a,b) called the {sf rational associahedron}. It is a pure
simplicial complex of dimension a-2, and its maximal faces are counted by the
{sf rational Catalan number} Cat(x)=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 Ass(a,b) is shellable and give nice product
formulas for its h-vector (the {sf rational Narayana numbers}) and f-vector
(the {sf rational Kirkman numbers}). We define Ass(a,b) via {sf rational
Dyck paths}: lattice paths from (0,0) to (b,a) staying above the line y =
frac{a}{b}x. We also use rational Dyck paths to define a rational
generalization of noncrossing perfect matchings of [2n]. In the case (a,b) =
(n, mn+1), our construction produces the noncrossing partitions of [(m+1)n] in
which each block has size m+1. | | Source: | arXiv, 1305.7286 | | Services: | Forum | Review | PDF | Favorites | |
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