37 research outputs found
On Noncrossing and nonnesting partitions of type D
We present an explicit bijection between noncrossing and nonnesting
partitions of Coxeter systems of type D which preserves openers, closers and
transients.Comment: 13 pages, 10 figures. A remark on a reference has been correcte
Boolean algebras and spectrum
For any Boolean algebra we compute the spectrum of its associated
undirected Hasse graph.Fundação para a Ciência e a Tecnologia Grant
SFRH/BPD/30471/2006; Dipartimento di Matematica, UniversitĂ di Genova, Italy
Enumerating Sn by associated transpositions and linear extensions of finite posets
AbstractWe define a family of statistics over the symmetric group Sn indexed by subsets of the transpositions, and we study the corresponding generating functions. We show that they have many interesting combinatorial properties. In particular we prove that any poset of size n corresponds to a subset of transpositions of Sn, and that the generating function of the corresponding statistic includes partial linear extensions of such a poset. We prove equidistribution results, and we explicitly compute the associated generating functions for several classes of subsets
Dominant Shi regions with a fixed separating wall: bijective enumeration
We present a purely combinatorial proof by means of an explicit bijection, of the exact number of dominant regions having as a separating wall the hyperplane associated to the longest root in the m-extended Shi hyperplane arrangement of type A and dimension n-1
On the noncommutative hypergeometric equation
Recently, J. A. Tirao [Proc. Nat. Acad. Sci. 100 (14) (2003), 8138–8141]
considered a matrix-valued analogue of the 2F1 GauĂź hypergeometric function and
showed that it is the unique solution of a matrix-valued hypergeometric equation
analytic at z = 0 with value I, the identity matrix, at z = 0. We give an independent
proof of Tirao’s result, extended to the slightly more general setting of
hypergeometric functions over an abstract unital Banach algebra. We provide a
similar (but more complicated-looking) result for a second type of noncommutative
2F1 Gauß hypergeometric function.CMUC; FWF Austrian Science Fund grant P17563–N13; FWF Austrian Science Fund grant P17563–N13; FWF, S960
On an index two subgroup of puzzle and Littlewood-Richardson tableau Z2 x S3-symmetries
We consider an action of the dihedral group Z2 Ă— S3 on Littlewood-
Richardson tableaux which carries a linear time action of a subgroup of index two.
This index two subgroup action on Knutson-Tao-Woodward puzzles is the group
generated by the puzzle mirror reflections with label swapping. One shows that, as
happens in puzzles, half of the twelve symmetries of Littlewood-Richardson coefficients
may also be exhibited on Littlewood-Richardson tableaux by surprisingly easy
maps. The other hidden half symmetries are given by a remaining generator which
enables to reduce those symmetries to the Sch¨utzenberger involution. Purbhoo
mosaics are used to map the action of the subgroup of index two on Littlewood-
Richardson tableaux into the group generated by the puzzle mirror reflections with
label swapping. After Pak and Vallejo one knows that Berenstein-Zelevinsky triangles,
Knutson-Tao hives and Littlewood-Richardson tableaux may be put in correspondence
by linear algebraic maps. We conclude that, regarding the symmetries,
the behaviour of the various combinatorial models for Littlewood-Richardson coefficients
is similar, and the bijections exhibiting them are in a certain sense unique
Linear time equivalence of Littlewood--Richardson coefficient symmetry maps
Benkart, Sottile, and Stroomer have completely characterized by Knuth and
dual Knuth equivalence a bijective proof of the conjugation symmetry of the
Littlewood-Richardson coefficients. Tableau-switching provides an algorithm to
produce such a bijective proof. Fulton has shown that the White and the
Hanlon-Sundaram maps are versions of that bijection. In this paper one exhibits
explicitly the Yamanouchi word produced by that conjugation symmetry map which
on its turn leads to a new and very natural version of the same map already
considered independently. A consequence of this latter construction is that
using notions of Relative Computational Complexity we are allowed to show that
this conjugation symmetry map is linear time reducible to the Schutzenberger
involution and reciprocally. Thus the Benkart-Sottile-Stroomer conjugation
symmetry map with the two mentioned versions, the three versions of the
commutative symmetry map, and Schutzenberger involution, are linear time
reducible to each other. This answers a question posed by Pak and Vallejo.Comment: accepted in Discrete Mathematics and Theoretical Computer Science
Proceeding
Character Sums over Integers with Restricted g-ary Digits
This is a preprint of an article published in the Illinois Journal of Mathematics, 46 (2002) no.3, pp.819-836.We establish upper bounds for multiplicative character sums and exponential sums over sets of integers that are described by various
properties of their digits in a fixed base g ≥ 2. Our main tools are the Weil and Vinogradov bounds for character sums and exponential
sums. Our results can be applied to study the distribution of quadratic non-residues and primitive roots among these sets of integers
Number Theoretic Designs for Directed Regular Graphs of Small Diameter
©2004 Society for Industrial and Applied Mathematics.In 1989, F. R. K. Chung gave a construction for certain directed h-regular graphs of small diameter. Her construction is based on finite fields, and the upper bound on the diameter
of these graphs is derived from bounds for certain very short character sums. Here we present two similar constructions that are based on properties of discrete logarithms and exponential functions in residue rings modulo a prime power. Accordingly, we use bounds for certain sums with additive and multiplicative characters to estimate the diameter of our graphs. We also give a third construction
that avoids the use of bounds for exponential sums