An extension of Tamari lattices

For any finite path $v$ on the square grid consisting of north and east unit steps, starting at (0,0), we construct a poset Tam$(v)$ that consists of all the paths weakly above $v$ with the same number of north and east steps as $v$. For particular choices of $v$, we recover the traditional Tamari lattice and the $m$-Tamari lattice. Let $\overleftarrow{v}$ be the path obtained from $v$ by reading the unit steps of $v$ in reverse order, replacing the east steps by north steps and vice versa. We show that the poset Tam$(v)$ is isomorphic to the dual of the poset Tam$(\overleftarrow{v})$. We do so by showing bijectively that the poset Tam$(v)$ is isomorphic to the poset based on rotation of full binary trees with the fixed canopy $v$, from which the duality follows easily. This also shows that Tam$(v)$ is a lattice for any path $v$. We also obtain as a corollary of this bijection that the usual Tamari lattice, based on Dyck paths of height $n$, is a partition of the (smaller) lattices Tam$(v)$, where the $v$ are all the paths on the square grid that consist of $n-1$ unit steps. We explain possible connections between the poset Tam$(v)$ and (the combinatorics of) the generalized diagonal coinvariant spaces of the symmetric group.Comment: 18 page

Rhombic alternative tableaux, assemblees of permutations, and the ASEP

International audienceIn this paper, we introduce therhombic alternative tableaux, whose weight generating functions providecombinatorial formulae to compute the steady state probabilities of the two-species ASEP. In the ASEP, there aretwo species of particles, oneheavyand onelight, on a one-dimensional finite lattice with open boundaries, and theparametersα,β, andqdescribe the hopping probabilities. The rhombic alternative tableaux are enumerated by theLah numbers, which also enumerate certainassembl ́ees of permutations. We describe a bijection between the rhombicalternative tableaux and these assembl ́ees. We also provide an insertion algorithm that gives a weight generatingfunction for the assemb ́ees. Combined, these results give a bijective proof for the weight generating function for therhombic alternative tableaux

Vicious walkers, friendly walkers and Young tableaux II: With a wall

We derive new results for the number of star and watermelon configurations of vicious walkers in the presence of an impenetrable wall by showing that these follow from standard results in the theory of Young tableaux, and combinatorial descriptions of symmetric functions. For the problem of $n$-friendly walkers, we derive exact asymptotics for the number of stars and watermelons both in the absence of a wall and in the presence of a wall.Comment: 35 pages, AmS-LaTeX; Definitions of n-friendly walkers clarified; the statement of Theorem 4 and its proof were correcte

Formal Power Series and Algebraic Combinatorics

Catalan tableaux and the asymmetric exclusion proces

doi:10.1088/1742-6596/42/1/024 Counting Complexity Multi-directed animals, connected heaps of dimers and Lorentzian triangulations 1

Abstract. Bousquet-Mélou and Rechnitzer have introduced the class of multi-directed animals, as an extension of the classical 2D directed animals. They gave an explicit expression for their generating function. In the case of directed animals, the corresponding generating function is algebraic and various “combinatorial explanations ” (bijective proofs) have been given, in particular using the so-called model of heaps of dimers. Although the generating function for multi-directed animals is not algebraic, even worst not D-finite, we give a bijective proof of Bousquet-Mélou and Rechnitzer’s formula, introducing the “Nordic decomposition ” of a connected heap of dimers. One possible interest of this bijective proof is in relation with 2D Lorentzian quantum gravity. Ambjørn, Loll, Di Francesco, Guitter and Kristjansen have introduced and studied the notion of Lorentzian triangulations. There exist correspondences between these triangulations, connected heaps of dimers and multi-directed animals