Abstract. We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of [n], as well as over all matchings on [2n]. As a corollary, the number of knoncrossing partitions is equal to the number of k-nonnesting partitions. The same is also true for matchings. An application is given to the enumeration of matchings with no k-crossing (or with no k-nesting).
Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature)×(center) is an integer vector. This series of papers explain such properties.A Descartes configuration is a set of four mutually tangent circles with disjoint interiors. An Apollonian circle packing can be described in terms of the Descartes configuration it contains. We describe the space of all ordered, oriented Descartes configurations using a coordinate system M D consisting of those 4 × 4 real matrices W withOn the parameter * Ronald L.space M D the group Aut(Q D ) acts on the left, and Aut(Q W ) acts on the right, giving two different "geometric" actions. Both these groups are isomorphic to the Lorentz group O (3, 1). The right action of Aut(Q W ) (essentially) corresponds to Möbius transformations acting on the underlying Euclidean space R 2 while the left action of Aut(Q D ) is defined only on the parameter space. We observe that the Descartes configurations in each Apollonian packing form an orbit of a single Descartes configuration under a certain finitely generated discrete subgroup of Aut(Q D ), which we call the Apollonian group. This group consists of 4×4 integer matrices, and its integrality properties lead to the integrality properties observed in some Apollonian circle packings.We introduce two more related finitely generated groups in Aut(Q D ), the dual Apollonian group produced from the Apollonian group by a "duality" conjugation, and the superApollonian group which is the group generated by the Apollonian and dual Apollonian groups together. These groups also consist of integer 4 × 4 matrices. We show these groups are hyperbolic Coxeter groups.
A generalized x-parking function associated to a positive integer vector of the form a b b b is a sequence a 1 a 2 a n of positive integers whose nondecreasing rearrangementThe set of x-parking functions has the same cardinality as the set of sequences of rooted b-forests on n . We construct a bijection between these two sets. We show that the sum enumerator of complements of x-parking functions is identical to the inversion enumerator of sequences of rooted b-forests by generating function analysis. Combinatorial correspondences between the sequences of rooted forests and x-parking functions are also given in terms of depth-first search and breadth-first search on multicolored graphs. 2001 Elsevier Science
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