Distinguished Visiting Professor Talk Series: Sergi Elizalde, Dartmouth College, November 24th @4pm in Olin 372

Title: Consecutive patterns in permutations


Abstract: An occurrence of a consecutive pattern sigma in a permutation pi
is a subsequence of adjacent entries of pi in the same relative
order as the entries of sigma. For example, occurrences of the
consecutive pattern 21 are descents, and alternating permutations
are those that avoid the consecutive patterns 123 and 321. The
systematic study of consecutive patterns in permutations started over
a decade ago.  More recently, consecutive patterns have become
relevant in the study of one-dimensional dynamical systems, and they
are useful in creating tests to distinguish random from deterministic
time series.

We show that the number of permutations avoiding the consecutive
pattern 12…m —that is, containing no m adjacent entries in
increasing order— is asymptotically larger than the number of
permutations avoiding any other consecutive pattern of length m. At
the other end of the spectrum, the number of permutations avoiding
12… (m-2)m(m-1) is asymptotically smaller than for any other
consecutive pattern.

Student Talk Series: Sergi Elizalde, Dartmouth College, November 20th, Olin 268 @ noon

Title: Lattice paths between two boundaries

Abstract: Dyck paths are lattice paths with steps N=(0,1) and E=(0,1) starting
at (0,0), ending at (n,n), and never going below the diagonal y=x.
Among the many interesting facts known about Dyck paths, one is that
the parameters ‘number of E steps at the end’ and ‘number of returns
to the diagonal’ have a symmetric joint distribution, meaning that to
each Dyck path we can bijectively associate another one where these
parameters are interchanged.

I will show that this symmetry property applies not only to Dyck
paths, but more generally to lattice paths with N and E steps that lie
between any two fixed boundaries. Finally, I will discuss how these
lattice paths are related to other objects in combinatorics, such as
matroids, semistandard Young tableaux and k-triangulations.

This is joint work with Martin Rubey.