6 research outputs found
Pattern-functions, statistics, and shallow permutations
We study relationships between permutation statistics and pattern-functions,
counting the number of times particular patterns occur in a permutation. This
allows us to write several familiar statistics as linear combinations of
pattern counts, both in terms of a permutation and in terms of its image under
the fundamental bijection. We use these enumerations to resolve the question of
characterizing so-called "shallow" permutations, whose depth (equivalently,
disarray/displacement) is minimal with respect to length and reflection length.
We present this characterization in several ways, including vincular patterns,
mesh patterns, and a new object that we call "arrow patterns." Furthermore, we
specialize to characterizing and enumerating shallow involutions and shallow
cycles, encountering the Motzkin and large Schr\"oder numbers, respectively.Comment: 17 pages, to appear in The Electronic Journal of Combinatoric
Coincidence among families of mesh patterns
Two mesh patterns are coincident if they are avoided by the same set of permutations. In this paper, we provide necessary conditions for this coincidence, which include having the same set of enclosed diagonals. This condition is sufficient to prove coincidence of vincular patterns, although it is not enough to guarantee coincidence of bivincular patterns. In addition, we provide a generalization of the Shading Lemma (Hilmarsson et al.), a result that examined when a square could be added to the mesh of a pattern