Bounded Affine Permutations II. Avoidance of Decreasing Patterns

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.

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Affine Permutations of Type $A$

The Electronic Journal of Combinatorics ◽

10.37236/1276 ◽

1995 ◽

Vol 3 (2) ◽

Cited By ~ 15

Author(s):

Anders Björner ◽

Francesco Brenti

Keyword(s):

Coxeter Group ◽

Type A ◽

Bruhat Order ◽

Weak Order ◽

Combinatorial Properties ◽

Affine Permutations

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

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321-avoiding affine permutations, heaps, and periodic parallelogram polyominoes

Electronic Notes in Discrete Mathematics ◽

10.1016/j.endm.2017.05.009 ◽

2017 ◽

Vol 59 ◽

pp. 115-130

Author(s):

Riccardo Biagioli ◽

Frédéric Jouhet ◽

Philippe Nadeau

Keyword(s):

Affine Permutations

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Canonical Decompositions of Affine Permutations, Affine Codes, and Split $k$-Schur Functions

The Electronic Journal of Combinatorics ◽

10.37236/2248 ◽

2012 ◽

Vol 19 (4) ◽

Cited By ~ 5

Author(s):

Tom Denton

Keyword(s):

Schur Functions ◽

Canonical Decomposition ◽

Schur Function ◽

Combinatorial Properties ◽

Affine Permutations ◽

New Perspective ◽

Special Case ◽

Arbitrary Affine

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.

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