On expected factors in reduced decompositions in type B

AbstractThe expected number of Yang–Baxter moves appearing in a reduced decomposition of the longest element of the Coxeter group of type Bn is computed to be 2−4/n. For the same element, the expected number of 0101 or 1010 factors appearing in a reduced decomposition is 2/(n2−2)

Enumerations relating braid and commutation classes

We obtain an upper and lower bound for the number of reduced words for a permutation in terms of the number of braid classes and the number of commutation classes of the permutation. We classify the permutations that achieve each of these bounds, and enumerate both cases.Comment: 19 page

The combinatorics of reduced decompositions

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006.Includes bibliographical references (p. 85-88) and index.This thesis examines several aspects of reduced decompositions in finite Coxeter groups. Effort is primarily concentrated on the symmetric group, although some discussions are subsequently expanded to finite Coxeter groups of types B and D. In the symmetric group, the combined frameworks of permutation patterns and reduced decompositions are used to prove a new characterization of vexillary permutations. This characterization and the methods used yield a variety of new results about the structure of several objects relating to a permutation. These include its commutation classes, the corresponding graph of the classes, the zonotopal tilings of a particular polygon, and a poset defined in terms of these tilings. The class of freely braided permutations behaves particularly well, and its graphs and posets are explicitly determined. The Bruhat order for the symmetric group is examined, and the permutations with boolean principal order ideals are completely characterized. These form an order ideal which is a simplicial poset, and its rank generating function is computed. Moreover, it is determined when the set of permutations avoiding a particular set of patterns is an order ideal, and the rank generating functions of these ideals are computed.(cont.) The structure of the intervals and order ideals in this poset is elucidated via patterns, including progress towards understanding the relationship between pattern containment and subintervals in principal order ideals. The final discussions of the thesis are on reduced decompositions in the finite Coxeter groups of types B and D. Reduced decompositions of the longest element in the hyperoctahedral group are studied, and expected values are calculated, expanding on previous work for the symmetric group. These expected values give a quantitative interpretation of the effects of the Coxeter relations on reduced decompositions of the longest element in this group. Finally, the Bruhat order in types B and D is studied, and the elements in these groups with boolean principal order ideals are characterized and enumerated by length.by Bridget Eileen Tenner.Ph.D