2,200 research outputs found
The sorting index
We consider a bivariate polynomial that generalizes both the length and
reflection length generating functions in a finite Coxeter group. In seeking a
combinatorial description of the coefficients, we are led to the study of a new
Mahonian statistic, which we call the sorting index. The sorting index of a
permutation and its type B and type D analogues have natural combinatorial
descriptions which we describe in detail.Comment: 14 pages, minor changes, new references adde
Ordering Metro Lines by Block Crossings
A problem that arises in drawings of transportation networks is to minimize
the number of crossings between different transportation lines. While this can
be done efficiently under specific constraints, not all solutions are visually
equivalent. We suggest merging crossings into block crossings, that is,
crossings of two neighboring groups of consecutive lines. Unfortunately,
minimizing the total number of block crossings is NP-hard even for very simple
graphs. We give approximation algorithms for special classes of graphs and an
asymptotically worst-case optimal algorithm for block crossings on general
graphs. That is, we bound the number of block crossings that our algorithm
needs and construct worst-case instances on which the number of block crossings
that is necessary in any solution is asymptotically the same as our bound
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