18,302 research outputs found
Pattern avoidance in forests of binary shrubs
We investigate pattern avoidance in permutations satisfying some additional restrictions. These are naturally considered in terms of avoiding patterns in linear extensions of certain forest-like partially ordered sets, which we call binary shrub forests. In this context, we enumerate forests avoiding patterns of length three. In four of the five non-equivalent cases, we present explicit enumerations by exhibiting bijections with certain lattice paths bounded above by the line y = lx, for some l in Q+, one of these being the celebrated Duchonās club paths with l = 2/3. In the remaining case, we use the machinery of analytic combinatorics to determine the minimal polynomial of its generating function, and deduce its growth rate
On Buffon Machines and Numbers
The well-know needle experiment of Buffon can be regarded as an analog (i.e.,
continuous) device that stochastically "computes" the number 2/pi ~ 0.63661,
which is the experiment's probability of success. Generalizing the experiment
and simplifying the computational framework, we consider probability
distributions, which can be produced perfectly, from a discrete source of
unbiased coin flips. We describe and analyse a few simple Buffon machines that
generate geometric, Poisson, and logarithmic-series distributions. We provide
human-accessible Buffon machines, which require a dozen coin flips or less, on
average, and produce experiments whose probabilities of success are expressible
in terms of numbers such as, exp(-1), log 2, sqrt(3), cos(1/4), aeta(5).
Generally, we develop a collection of constructions based on simple
probabilistic mechanisms that enable one to design Buffon experiments involving
compositions of exponentials and logarithms, polylogarithms, direct and inverse
trigonometric functions, algebraic and hypergeometric functions, as well as
functions defined by integrals, such as the Gaussian error function.Comment: Largely revised version with references and figures added. 12 pages.
In ACM-SIAM Symposium on Discrete Algorithms (SODA'2011
Generating all permutations by context-free grammars in Chomsky normal form
Let Ln be the finite language of all n! strings that are permutations of n different symbols (n1). We consider context-free grammars Gn in Chomsky normal form that generate Ln. In particular we study a few families {Gn}n1, satisfying L(Gn)=Ln for n1, with respect to their descriptional complexity, i.e. we determine the number of nonterminal symbols and the number of production rules of Gn as functions of n
Pattern Avoidance in Task-Precedence Posets
We have extended classical pattern avoidance to a new structure: multiple
task-precedence posets whose Hasse diagrams have three levels, which we will
call diamonds. The vertices of each diamond are assigned labels which are
compatible with the poset. A corresponding permutation is formed by reading
these labels by increasing levels, and then from left to right. We used Sage to
form enumerative conjectures for the associated permutations avoiding
collections of patterns of length three, which we then proved. We have
discovered a bijection between diamonds avoiding 132 and certain generalized
Dyck paths. We have also found the generating function for descents, and
therefore the number of avoiders, in these permutations for the majority of
collections of patterns of length three. An interesting application of this
work (and the motivating example) can be found when task-precedence posets
represent warehouse package fulfillment by robots, in which case avoidance of
both 231 and 321 ensures we never stack two heavier packages on top of a
lighter package.Comment: 17 page
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