Enumeration of pyramids of one-dimensional pieces of arbitrary fixed integer length

We consider pyramids made of one-dimensional pieces of fixed integer length a and which may have pairwise overlaps of integer length from 1 to a. We prove that the number of pyramids of size m, i.e. consisting of m pieces, equals (am-1,m-1) for each a >= 2. This generalises a well known result for a = 2. A bijective correspondence between so-called right (or left) pyramids and a-ary trees is pointed out, and it is shown that asymptotically the average width of pyramids is proportional to the square root of the size

A note on the enumeration of directed animals via gas considerations

In the literature, most of the results about the enumeration of directed animals on lattices via gas considerations are obtained by a formal passage to the limit of enumeration of directed animals on cyclical versions of the lattice. Here we provide a new point of view on this phenomenon. Using the gas construction given in [Electron. J. Combin. (2007) 14 R71], we describe the gas process on the cyclical versions of the lattices as a cyclical Markov chain (roughly speaking, Markov chains conditioned to come back to their starting point). Then we introduce a notion of convergence of graphs, such that if $(G_n)\to G$ then the gas process built on $G_n$ converges in distribution to the gas process on $G$. That gives a general tool to show that gas processes related to animals enumeration are often Markovian on lines extracted from lattices. We provide examples and computations of new generating functions for directed animals with various sources on the triangular lattice, on the $\mathcal {T}_n$ lattices introduced in [Ann. Comb. 4 (2000) 269--284] and on a generalization of the \mathcaligr {L}_n lattices introduced in [J. Phys. A 29 (1996) 3357--3365].Comment: Published in at http://dx.doi.org/10.1214/08-AAP580 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Garside combinatorics for Thompson's monoid F+F^+ and a hybrid with the braid monoid B_∞+B\_\infty^+

On the model of simple braids, defined to be the left divisors of Garside's elements $\Delta\_n$ in the monoid $B\_\infty^+$ , we investigate simple elements in Thompson's monoid $F^+$ and in a larger monoid $H^+$ that is a hybrid of $B\_\infty^+$ and $F^+$ : in both cases, we count how many simple elements left divide the right lcm of the first n -- 1 atoms, and characterize their normal forms in terms of forbidden factors. In the case of $H^+$, a generalized Pascal triangle appears

The local limit of rooted directed animals on the square lattice

We consider the local limit of finite uniformly distributed directed animals on the square lattice viewed from the root. Two constructions of the resulting uniform infinite directed animal are given: one as a heap of dominoes, constructed by letting gravity act on a right-continuous random walk and one as a Markov process, obtained by slicing the animal horizontally. We look at geometric properties of this local limit and prove, in particular, that it consists of a single vertex at infinitely many (random) levels. Several martingales are found in connection with the confinement of the infinite directed animal on the non-negative coordinates.Comment: 59 pages, 16 figure