Résumé | A permutation class is a downset in the subpermutation ordering. Permutation classes arise naturally in a variety of fields, such as the study of Shubert varieties, Chebyshev polynomials, and ordering algoritms. A permutation class C may be defined the set of minimal elements not belonging to it, its basis B. In his Ph.D. thesis, Olivier Guibert introduced eleven different permutation classes, whose basis contains four permutations of length four, each one enumerated by the central binomial coefficients. It is well known that the binomial coefficients enumerate also the set of Dyck prefixes of a given length.
In the present talk, we define a bijection between the permutation class of basis B={3241,3214,4213,4231} and the set of Dyck prefixes. This bijection allows us to relate particular features of a Dyck prefix with the occurrences of some statistics in the corresponding permutation (ascents, left-to-right maxima, length of the greatest decreasing sequence, etc). Moreover, we try to find similar bijections for the other ten classes enumerated by the central binomial coefficients, in order to find generating function for the distribution of some permutation statistics over these classes. |