1,626 research outputs found
Clusters, generating functions and asymptotics for consecutive patterns in permutations
We use the cluster method to enumerate permutations avoiding consecutive
patterns. We reprove and generalize in a unified way several known results and
obtain new ones, including some patterns of length 4 and 5, as well as some
infinite families of patterns of a given shape. By enumerating linear
extensions of certain posets, we find a differential equation satisfied by the
inverse of the exponential generating function counting occurrences of the
pattern. We prove that for a large class of patterns, this inverse is always an
entire function. We also complete the classification of consecutive patterns of
length up to 6 into equivalence classes, proving a conjecture of Nakamura.
Finally, we show that the monotone pattern asymptotically dominates (in the
sense that it is easiest to avoid) all non-overlapping patterns of the same
length, thus proving a conjecture of Elizalde and Noy for a positive fraction
of all patterns
Mathematical practice, crowdsourcing, and social machines
The highest level of mathematics has traditionally been seen as a solitary
endeavour, to produce a proof for review and acceptance by research peers.
Mathematics is now at a remarkable inflexion point, with new technology
radically extending the power and limits of individuals. Crowdsourcing pulls
together diverse experts to solve problems; symbolic computation tackles huge
routine calculations; and computers check proofs too long and complicated for
humans to comprehend.
Mathematical practice is an emerging interdisciplinary field which draws on
philosophy and social science to understand how mathematics is produced. Online
mathematical activity provides a novel and rich source of data for empirical
investigation of mathematical practice - for example the community question
answering system {\it mathoverflow} contains around 40,000 mathematical
conversations, and {\it polymath} collaborations provide transcripts of the
process of discovering proofs. Our preliminary investigations have demonstrated
the importance of "soft" aspects such as analogy and creativity, alongside
deduction and proof, in the production of mathematics, and have given us new
ways to think about the roles of people and machines in creating new
mathematical knowledge. We discuss further investigation of these resources and
what it might reveal.
Crowdsourced mathematical activity is an example of a "social machine", a new
paradigm, identified by Berners-Lee, for viewing a combination of people and
computers as a single problem-solving entity, and the subject of major
international research endeavours. We outline a future research agenda for
mathematics social machines, a combination of people, computers, and
mathematical archives to create and apply mathematics, with the potential to
change the way people do mathematics, and to transform the reach, pace, and
impact of mathematics research.Comment: To appear, Springer LNCS, Proceedings of Conferences on Intelligent
Computer Mathematics, CICM 2013, July 2013 Bath, U
Quasipolynomial formulas for the Kronecker coefficients indexed by two two-row shapes (extended abstract)
We show that the Kronecker coefficients indexed by two two–row shapes are given
by quadratic quasipolynomial formulas whose domains are the maximal cells of a fan. Simple
calculations provide explicitly the quasipolynomial formulas and a description of the associated
fan.
These new formulas are obtained from analogous formulas for the corresponding reduced
Kronecker coefficients and a formula recovering the Kronecker coefficients from the reduced
Kronecker coefficients.
As an application, we characterize all the Kronecker coefficients indexed by two two-row
shapes that are equal to zero. This allowed us to disprove a conjecture of Mulmuley about the
behavior of the stretching functions attached to the Kronecker coefficients.Ministerio de Educación y Ciencia MTM2007–64509Junta de AndalucÃa FQM–33
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