372 research outputs found
Trivial Meet and Join within the Lattice of Monotone Triangles
The lattice of monotone triangles ordered by
entry-wise comparisons is studied. Let denote the unique minimal
element in this lattice, and the unique maximum. The number of
-tuples of monotone triangles with minimal infimum
(maximal supremum , resp.) is shown to
asymptotically approach as . Thus, with
high probability this event implies that one of the is
(, resp.). Higher-order error terms are also discussed.Comment: 15 page
The many faces of alternating-sign matrices
I give a survey of different combinatorial forms of alternating-sign
matrices, starting with the original form introduced by Mills, Robbins and
Rumsey as well as corner-sum matrices, height-function matrices,
three-colorings, monotone triangles, tetrahedral order ideals, square ice,
gasket-and-basket tilings and full packings of loops.Comment: 22 pages, 16 figures; presented at "Discrete Models" conference
(Paris, July 2001
A Recipe for State-and-Effect Triangles
In the semantics of programming languages one can view programs as state
transformers, or as predicate transformers. Recently the author has introduced
state-and-effect triangles which capture this situation categorically,
involving an adjunction between state- and predicate-transformers. The current
paper exploits a classical result in category theory, part of Jon Beck's
monadicity theorem, to systematically construct such a state-and-effect
triangle from an adjunction. The power of this construction is illustrated in
many examples, covering many monads occurring in program semantics, including
(probabilistic) power domains
Generating Random Elements of Finite Distributive Lattices
This survey article describes a method for choosing uniformly at random from
any finite set whose objects can be viewed as constituting a distributive
lattice. The method is based on ideas of the author and David Wilson for using
``coupling from the past'' to remove initialization bias from Monte Carlo
randomization. The article describes several applications to specific kinds of
combinatorial objects such as tilings, constrained lattice paths, and
alternating-sign matrices.Comment: 13 page
On comparability of bigrassmannian permutations
Let Sn and Gn denote the respective sets of ordinary and bigrassmannian (BG) permutations of order n, and let (Gn,≤) denote the Bruhat ordering permutation poset. We study the restricted poset (Bn,≤), first providing a simple criterion for comparability. This criterion is used to show that that the poset is connected, to enumerate the saturated chains between elements, and to enumerate the number of maximal elements below r fixed elements. It also quickly produces formulas for β(ω) (α(ω), respectively), the number of BG permutations weakly below (weakly above, respectively) a fixed ω ∈ Bn, and is used to compute the Mo¨bius function on any interval in Bn.
We then turn to a probabilistic study of β = β(ω) (α = α(ω) respectively) for the uniformly random ω ∈ Bn. We show that α and β are equidistributed, and that β is of the same order as its expectation with high probability, but fails to concentrate about its mean. This latter fact derives from the limiting distribution of β/n3. We also compute the probability that randomly chosen BG permutations form a 2- or 3-element multichain
Uniqueness for the signature of a path of bounded variation and the reduced path group
We introduce the notions of tree-like path and tree-like equivalence between
paths and prove that the latter is an equivalence relation for paths of finite
length. We show that the equivalence classes form a group with some similarity
to a free group, and that in each class there is one special tree reduced path.
The set of these paths is the Reduced Path Group. It is a continuous analogue
to the group of reduced words. The signature of the path is a power series
whose coefficients are definite iterated integrals of the path. We identify the
paths with trivial signature as the tree-like paths, and prove that two paths
are in tree-like equivalence if and only if they have the same signature. In
this way, we extend Chen's theorems on the uniqueness of the sequence of
iterated integrals associated with a piecewise regular path to finite length
paths and identify the appropriate extended meaning for reparameterisation in
the general setting. It is suggestive to think of this result as a
non-commutative analogue of the result that integrable functions on the circle
are determined, up to Lebesgue null sets, by their Fourier coefficients. As a
second theme we give quantitative versions of Chen's theorem in the case of
lattice paths and paths with continuous derivative, and as a corollary derive
results on the triviality of exponential products in the tensor algebra.Comment: 52 pages - considerably extended and revised version of the previous
version of the pape
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