6 research outputs found
Simple formulas for lattice paths avoiding certain periodic staircase boundaries
There is a strikingly simple classical formula for the number of lattice
paths avoiding the line x = ky when k is a positive integer. We show that the
natural generalization of this simple formula continues to hold when the line x
= ky is replaced by certain periodic staircase boundaries--but only under
special conditions. The simple formula fails in general, and it remains an open
question to what extent our results can be further generalized.Comment: Accepted version (JCTA); proof of Corollary 7 expanded, and 2 new
refs adde
Two operators on sandpile configurations, the sandpile model on the complete bipartite graph, and a Cyclic Lemma
We introduce two operators on stable configurations of the sandpile model
that provide an algorithmic bijection between recurrent and parking
configurations. This bijection preserves their equivalence classes with respect
to the sandpile group. The study of these operators in the special case of the
complete bipartite graph naturally leads to a generalization of the
well known Cyclic Lemma of Dvoretsky and Motzkin, via pairs of periodic
bi-infinite paths in the plane having slightly different slopes. We achieve our
results by interpreting the action of these operators as an action on a point
in the grid which is pointed to by one of these pairs of paths.
Our Cyclic lemma allows us to enumerate several classes of polyominoes, and
therefore builds on the work of Irving and Rattan (2009), Chapman et al.
(2009), and Bonin et al. (2003).Comment: 28 page
Symmetries of statistics on lattice paths between two boundaries
We prove that on the set of lattice paths with steps N=(0,1) and E=(1,0) that
lie between two fixed boundaries T and B (which are themselves lattice paths),
the statistics `number of E steps shared with B' and `number of E steps shared
with T' have a symmetric joint distribution. To do so, we give an involution
that switches these statistics, preserves additional parameters, and
generalizes to paths that contain steps S=(0,-1) at prescribed x-coordinates.
We also show that a similar equidistribution result for path statistics follows
from the fact that the Tutte polynomial of a matroid is independent of the
order of its ground set. We extend the two theorems to k-tuples of paths
between two boundaries, and we give some applications to Dyck paths,
generalizing a result of Deutsch, to watermelon configurations, to
pattern-avoiding permutations, and to the generalized Tamari lattice. Finally,
we prove a conjecture of Nicol\'as about the distribution of degrees of k
consecutive vertices in k-triangulations of a convex n-gon. To achieve this
goal, we provide a new statistic-preserving bijection between certain k-tuples
of non-crossing paths and k-flagged semistandard Young tableaux, which is based
on local moves reminiscent of jeu de taquin.Comment: Small typos corrected, and journal reference and grant info adde