scholarly journals The enumeration of fully commutative affine permutations

2010 ◽  
Vol 31 (5) ◽  
pp. 1342-1359 ◽  
Author(s):  
Christopher R.H. Hanusa ◽  
Brant C. Jones
Keyword(s):  
2019 ◽  
Vol 162 ◽  
pp. 271-305
Author(s):  
Riccardo Biagioli ◽  
Frédéric Jouhet ◽  
Philippe Nadeau
Keyword(s):  

2011 ◽  
Vol 46 (1-4) ◽  
pp. 175-191 ◽  
Author(s):  
Eric Clark ◽  
Richard Ehrenborg
Keyword(s):  

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Christopher R. H. Hanusa ◽  
Brant C. Jones

International audience We give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length, extending formulas due to Stembridge and Barcucci–Del Lungo–Pergola–Pinzani. For fixed rank, the length generating functions have coefficients that are periodic with period dividing the rank. In the course of proving these formulas, we obtain results that elucidate the structure of the fully commutative affine permutations. This is a summary of the results; the full version appears elsewhere. Nous présentons une fonction génératrice qui énumère les permutations affines totalement commutatives par leur rang et par leur longueur de Coxeter, généralisant les formules dues à Stembridge et à Barcucci–Del Lungo–Pergola–Pinzani. Pour un rang précis, les fonctions génératrices ont des coefficients qui sont périodiques de période divisant leur rang. Nous obtenons des résultats qui expliquent la structure des permutations affines totalement commutatives. L'article dessous est un aperçu des résultats; la version complète appara\^ıt ailleurs.


10.37236/1276 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Anders Björner ◽  
Francesco Brenti

We study combinatorial properties, such as inversion table, weak order and Bruhat order, for certain infinite permutations that realize the affine Coxeter group $\tilde{A}_{n}$.


2017 ◽  
Vol 59 ◽  
pp. 115-130
Author(s):  
Riccardo Biagioli ◽  
Frédéric Jouhet ◽  
Philippe Nadeau
Keyword(s):  

10.37236/2248 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Tom Denton

We develop a new perspective on the unique maximal decomposition of an arbitrary affine permutation into a product of cyclically decreasing elements, implicit in work of Thomas Lam.  This decomposition is closely related to the affine code, which generalizes the $k$-bounded partition associated to Grassmannian elements.  We also prove that the affine code readily encodes a number of basic combinatorial properties of an affine permutation.  As an application, we prove a new special case of the Littlewood-Richardson Rule for $k$-Schur functions, using the canonical decomposition to control for which permutations appear in the expansion of the $k$-Schur function in noncommuting variables over the affine nil-Coxeter algebra.


10.37236/6176 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Guizhi Qin ◽  
Sherry H.F. Yan

As a variation of De Bruijn graphs on strings of symbols, the graph of overlapping permutations has a directed edge $\pi(1)\pi(2)\ldots \pi(n+1)$ from the standardization of $\pi(1)\pi(2)\ldots \pi(n)$ to the standardization of $\pi(2)\pi(3)\ldots \pi(n+1)$. In this paper, we consider the enumeration of $d$-cycles in the subgraph of overlapping $(231, 4\bar{1}32)$-avoiding permutations. To this end, we introduce the notions of marked Motzkin paths and marked Riordan paths, where a marked Motzkin (resp. Riordan) path is a Motzkin (resp. Riordan) path in which exactly one step before the leftmost return point is marked. We show that the number of closed walks of length $d$ in the subgraph of overlapping $(231, 4\bar{1}32)$-avoiding permutations are closely related to the number of marked Motzkin paths and that of marked Riordan paths.  By establishing bijections, we get the enumerations of marked Motzkin paths and marked Riordan paths. As a corollary, we provide bijective proofs of two identities involving Catalan numbers in answer to the problem posed by Ehrenborg, Kitaev and Steingrímsson. Moreover, we get the enumerations of $(231, 4\bar{1}32)$-avoiding affine permutations and $(312, 32\bar{4}1)$-avoiding affine permutations.


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