32 research outputs found
A second look at the toric h-polynomial of a cubical complex
We provide an explicit formula for the toric -contribution of each cubical
shelling component, and a new combinatorial model to prove Clara Chan's result
on the non-negativity of these contributions. Our model allows for a variant of
the Gessel-Shapiro result on the -polynomial of the cubical lattice, this
variant may be shown by simple inclusion-exclusion. We establish an isomorphism
between our model and Chan's model and provide a reinterpretation in terms of
noncrossing partitions. By discovering another variant of the Gessel-Shapiro
result in the work of Denise and Simion, we find evidence that the toric
-polynomials of cubes are related to the Morgan-Voyce polynomials via
Viennot's combinatorial theory of orthogonal polynomials.Comment: Minor correction
Tree expansions of some Lie idempotents}
We prove that the Catalan Lie idempotent , introduced in
[Menous {\it et al.}, Adv. Appl. Math. 51 (2013), 177] can be
refined by introducing independent parameters
and that the coefficient of each monomial is itself a Lie idempotent
in the descent algebra. These new idempotents are multiplicity-free
sums of subsets of the Poincar\'e-Birkhoff-Witt basis of the Lie module.
These results are obtained by embedding noncommutative symmetric functions
into the
dual noncommutative Connes-Kreimer algebra, which also allows us to
interpret, and rederive in a simpler way,
Chapoton's results on a two-parameter tree expanded series.Comment: 27 page
Functional programming and graph algorithms
This thesis is an investigation of graph algorithms in the non-strict purely functional language Haskell. Emphasis is placed on the importance of achieving an asymptotic complexity as good as with conventional languages. This is achieved by using the monadic model for including actions on the state. Work on the monadic model was carried out at Glasgow University by Wadler, Peyton Jones, and Launchbury in the early nineties and has opened up many diverse application areas. One area is the ability to express data structures that require sharing. Although graphs are not presented in this style, data structures that graph algorithms use are expressed in this style. Several examples of stateful algorithms are given including union/find for disjoint sets, and the linear time sort binsort.
The graph algorithms presented are not new, but are traditional algorithms recast in a functional setting. Examples include strongly connected components, biconnected components, Kruskal's minimum cost spanning tree, and Dijkstra's shortest paths. The presentation is lucid giving more insight than usual. The functional setting allows for complete calculational style correctness proofs - which is demonstrated with many examples.
The benefits of using a functional language for expressing graph algorithms are quantified by looking at the issues of execution times, asymptotic complexity, correctness, and clarity, in comparison with traditional approaches. The intention is to be as objective as possible, pointing out both the weaknesses and the strengths of using a functional language
The number of intervals in the m-Tamari lattices
An m-ballot path of size n is a path on the square grid consisting of north
and east steps, starting at (0,0), ending at (mn,n), and never going below the
line {x=my}. The set of these paths can be equipped with a lattice structure,
called the m-Tamari lattice, which generalizes the usual Tamari lattice
obtained when m=1. We prove that the number of intervals in this lattice is This formula was recently
conjectured by Bergeron in connection with the study of coinvariant spaces. The
case m=1 was proved a few years ago by Chapoton. Our proof is based on a
recursive description of intervals, which translates into a functional equation
satisfied by the associated generating function. The solution of this equation
is an algebraic series, obtained by a guess-and-check approach. Finding a
bijective proof remains an open problem.Comment: 19 page