589 research outputs found
Labelings for Decreasing Diagrams
This article is concerned with automating the decreasing diagrams technique
of van Oostrom for establishing confluence of term rewrite systems. We study
abstract criteria that allow to lexicographically combine labelings to show
local diagrams decreasing. This approach has two immediate benefits. First, it
allows to use labelings for linear rewrite systems also for left-linear ones,
provided some mild conditions are satisfied. Second, it admits an incremental
method for proving confluence which subsumes recent developments in automating
decreasing diagrams. The techniques proposed in the article have been
implemented and experimental results demonstrate how, e.g., the rule labeling
benefits from our contributions
Counting real rational functions with all real critical values
We study the number of real rational degree n functions (considered up to
linear fractional transformations of the independent variable) with a given set
of 2n-2 distinct real critical values. We present a combinatorial reformulation
of this number and pose several related questions.Comment: 12 pages (AMSTEX), 3 picture
The Rees product of posets
We determine how the flag f-vector of any graded poset changes under the Rees
product with the chain, and more generally, any t-ary tree. As a corollary, the
M\"obius function of the Rees product of any graded poset with the chain, and
more generally, the t-ary tree, is exactly the same as the Rees product of its
dual with the chain, respectively, t-ary chain. We then study enumerative and
homological properties of the Rees product of the cubical lattice with the
chain. We give a bijective proof that the M\"obius function of this poset can
be expressed as n times a signed derangement number. From this we derive a new
bijective proof of Jonsson's result that the M\"obius function of the Rees
product of the Boolean algebra with the chain is given by a derangement number.
Using poset homology techniques we find an explicit basis for the reduced
homology and determine a representation for the reduced homology of the order
complex of the Rees product of the cubical lattice with the chain over the
symmetric group.Comment: 21 pages, 1 figur
Schur times Schubert via the Fomin-Kirillov algebra
We study multiplication of any Schubert polynomial by a
Schur polynomial (the Schubert polynomial of a Grassmannian
permutation) and the expansion of this product in the ring of Schubert
polynomials. We derive explicit nonnegative combinatorial expressions for the
expansion coefficients for certain special partitions , including
hooks and the 2x2 box. We also prove combinatorially the existence of such
nonnegative expansion when the Young diagram of is a hook plus a box
at the (2,2) corner. We achieve this by evaluating Schubert polynomials at the
Dunkl elements of the Fomin-Kirillov algebra and proving special cases of the
nonnegativity conjecture of Fomin and Kirillov.
This approach works in the more general setup of the (small) quantum
cohomology ring of the complex flag manifold and the corresponding (3-point)
Gromov-Witten invariants. We provide an algebro-combinatorial proof of the
nonnegativity of the Gromov-Witten invariants in these cases, and present
combinatorial expressions for these coefficients
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