### Affine Permutations of Type A

"... D'edi'e `a D. Foata `a l'occasion de ses soixante bougies Abstract We study combinatorial properties, such as inversion table, weak order and Bruhat order, for certain infinite permutations that realize the affine Coxeter group ~An. ..."

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D'edi'e `a D. Foata `a l'occasion de ses soixante bougies Abstract We study combinatorial properties, such as inversion table, weak order and Bruhat order, for certain infinite permutations that realize the affine Coxeter group ~An.

### ❥ ❥ ❥ ❥ ❥ ❥

"... In this master’s thesis, I will discuss some aspects of four combinatorial games. These are Pegs (also known as solitaire), Pebbles, Pennies and Piles, which are described in this report. The main contents can be summarised as follows. • Already known results for Pegs, Pebbles and Piles, together wi ..."

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In this master’s thesis, I will discuss some aspects of four combinatorial games. These are Pegs (also known as solitaire), Pebbles, Pennies and Piles, which are described in this report. The main contents can be summarised as follows. • Already known results for Pegs, Pebbles and Piles, together with the standard techniques. • A study of Pennies, which is an entirely new game. On most boards, Pennies is not solvable. • The reachable area of Pegs in Z d, allowing diagonal moves, has been investigated. I have found that there still is a rather narrow limit on how far one can reach, but it is substantially larger than the limit obtained not allowing diagonal moves. • The solvability of Pegs on some unconventional boards has been investigated. I have also obtained some results on the solvability of a game using a kind of sideways Pegs move.

### The order dimension of Bruhat order on infinite Coxeter groups

, 2005

"... We give a quadratic lower bound and a cubic upper bound on the order dimension of the Bruhat (or strong) ordering of the affine Coxeter group Ãn. We also demonstrate that the order dimension of the Bruhat order is infinite for a large class of Coxeter groups. ..."

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We give a quadratic lower bound and a cubic upper bound on the order dimension of the Bruhat (or strong) ordering of the affine Coxeter group Ãn. We also demonstrate that the order dimension of the Bruhat order is infinite for a large class of Coxeter groups.

### THE FINITE ANTICHAIN PROPERTY IN COXETER GROUPS

"... Abstract. We prove that the weak order on an infinite Coxeter group contains infinite antichains if and only if the group is not affine. 1. ..."

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Abstract. We prove that the weak order on an infinite Coxeter group contains infinite antichains if and only if the group is not affine. 1.

### BIJECTIVE PROJECTIONS ON PARABOLIC QUOTIENTS OF AFFINE WEYL GROUPS

"... Abstract. Affine Weyl groups and their parabolic quotients are used extensively as indexing sets for objects in combinatorics, representation theory, algebraic geometry, and number theory. More-over, in the classical Lie types we can conveniently realize the elements of these quotients via intuitive ..."

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Abstract. Affine Weyl groups and their parabolic quotients are used extensively as indexing sets for objects in combinatorics, representation theory, algebraic geometry, and number theory. More-over, in the classical Lie types we can conveniently realize the elements of these quotients via intuitive geometric and combinatorial models such as abaci, alcoves, coroot lattice points, core partitions, and bounded partitions. In [1] Berg, Jones, and Vazirani described a bijection between n-cores with first part equal to k and (n − 1)-cores with first part less than or equal to k, and they interpret this bijection in terms of these other combinatorial models for the quotient of the affine symmetric group by the finite symmetric group. In this paper we discuss how to generalize the bijection of Berg-Jones-Vazirani to parabolic quotients of affine Weyl groups in type C. We develop techniques using the associated affine hyperplane arrangement to interpret this bijection geometri-cally as a projection of alcoves onto the hyperplane containing their coroot lattice points. We are thereby able to analyze this bijective projection in the language of various additional combinatorial models developed by Hanusa and Jones in [10], such as abaci, core partitions, and canonical reduced expressions in the Coxeter group. 1.

### f f

, 1994

"... The analysis of chessboard pebbling by Fan Chung, Ron Graham, John Morrison and Andrew Odlyzko is strengthened and generalized, rst to higher dimension and then to arbitrary posets. Subject Classication: Primary 05A15; secondary 05E99. 1 The pebbling game The pebbling game of Kontsevich is played o ..."

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The analysis of chessboard pebbling by Fan Chung, Ron Graham, John Morrison and Andrew Odlyzko is strengthened and generalized, rst to higher dimension and then to arbitrary posets. Subject Classication: Primary 05A15; secondary 05E99. 1 The pebbling game The pebbling game of Kontsevich is played on the grid points of the rst quadrant. One starts with a single pebble on the origin and a move consists of replacing any pebble with two pebbles, one above and one to the right of the vanishing pebble: f p p