Related papers: Chain lengths in the type $B$ Tamari lattice
The existence of greatest lower bounds in the imbalance order of path-length sequences of binary trees is seen to be a consequence of a joint monotonicity property of the greater and lower expension operations. Path length sequences that…
Alt $\nu$-Tamari lattices constitute a remarkable family of lattices associated with lattice paths that broadly generalize the Dyck and Tamari lattices. To systematically study the structural properties of this family, we introduce a…
In this paper, we exploit the combinatorics and geometry of triangulations of products of simplices to derive new results in the context of Catalan combinatorics of $\nu$-Tamari lattices. In our framework, the main role of "Catalan objects"…
We use a recently introduced combinatorial object, the interval-poset, to describe two bijections on intervals of the Tamari lattice. Both bijections give a combinatorial proof of some previously known results. The first one is an inner…
We characterize supersolvable lattices in terms of a certain modular type relation. McNamara and Thomas earlier characterized this class of lattices as those graded lattices having a maximal chain that consists of left-modular elements. Our…
Part B (of a project involving four Parts) is about "bases of lines", a concept introduced by C. Herrmann and the author in the late 80's. Bases of lines attempt to describe a given modular lattice in a geometric way akin to how projective…
I review our method to formulate chiral gauge theories on the lattice based on a two-cutoff lattice regularization, and discuss recent numerical results in a chiral U(1) gauge theory in 2D.
An extension of the Tamari lattice to the multiplihedra is discussed, along with projections to the composihedra and the Boolean lattice. The multiplihedra and composihedra are sequences of polytopes that arose in algebraic topology and…
In a tropical approximation, any tree-shaped line soliton solution, a member of the simplest class of soliton solutions of the Kadomtsev-Petviashvili (KP-II) equation, determines a chain of planar rooted binary trees, connected by right…
We establish one-to-one correspondences between maximal antichains in products of two finite linear orders and other mathematical objects, such as certain alignments of two strings, walks on a grid, lattice paths, words of two or three…
This work studies the relationship between the chains of an algebraic lattice and the order structure of the join-semilattice of its compact elements. The results are presented into four chapters, each corresponding to a paper written in…
We apply the method of dressing chains to reproduction of Toda lattice in the case of D=1 and D=2. On the example of modified equations $m_0TL$ and $m_1TL$ it is shown that combination of the Darboux and Schlesinger transformations results…
We enumerate the intervals in the Tamari lattices. For this, we introduce an inductive description of the intervals. Then a notion of "new interval" is defined and these are also enumerated. A a side result, the inverse of two special…
The KP-II equation possesses a class of line soliton solutions which can be qualitatively described via a tropical approximation as a chain of rooted binary trees, except at "critical" events where a transition to a different rooted binary…
We introduce new objects, the interval-posets, that encode intervals of the Tamari lattice. We then find a combinatorial interpretation of the bilinear form that appears in the functional equation of Tamari intervals described by Chapoton.…
To every partial order P, one associates a polynomial $\mathbb{D}_P$ in 4 variables that enumerates the intervals of P according to 4 parameters. Some symmetry properties of this polynomial are obtained for a specific family of posets, the…
The poset of maximal tubings of a graph generalizes several well-known and remarkable partial orders. Notable examples include the weak Bruhat order and the Tamari lattice, posets of maximal tubings for the complete graph and the path…
The Cambrian lattices, introduced in (Reading, 2006), generalize the Tamari lattice to any choice of Coxeter element in any finite Coxeter group. They are further generalized to the m-Cambrian lattices (Stump, Thomas, Williams, 2015).…
Lattice artefacts are used, through modified lattice actions, as a tool to find the largest instantons in a toroidal geometry [0,L]^3X[0,T] for T to infinity. It is conjectured that the largest instanton is associated with tunnelling…
Extending a proposal of Defant and Kravitz [Discrete Mathematics, \textbf{1}, 347 (2024)], we define Hitomezashi patterns and loops on a torus and provide several structural results for such loops. For a given pattern, our main theorems…