Related papers: Bijections and congruences involving lattice paths…
For a connected semisimple Lie group $G$ we describe an explicit collection of correspondences between the admissible dual of $G$ and the admissible dual of the Cartan motion group associated with $G$. We conjecture that each of these…
Inspired by a new mathematical model for bobbin lace, this paper considers finite lattice paths formed from the set of step vectors $\mathfrak{A}=$$\{\rightarrow,$ $\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$ with the restriction that…
We resolve the explicit bijection problem between symmetric plane partitions (SPPs) and quasi transpose complementary plane partitions (QTCPPs), introduced by Schreier-Aigner, who proved their equinumerosity. First, we relate this problem…
We categorise coherent band (aka nullification) pathways between knots and 2-component links. Additionally, we characterise the minimal coherent band pathways (with intermediates) between any two knots or 2-component links with small…
We investigate the asymptotic behavior of random paths on a graded graph which describes the subword order for words in two letters. This graph, denoted by $\mathcal{Z}$, has been introduced by Viennot, who also discovered a remarkable…
Two subfamilies of Motzkin paths, with the same numbers of up, down, horizontal steps were known to be equinumerous with ternary trees and related objects. We construct a bijection between these two families that does not use any auxiliary…
In the Stanley lattice defined on Dyck paths of size $n$, cover relations are obtained by replacing a valley $DU$ by a peak $UD$. We investigate a greedy version of this lattice, first introduced by Chenevi\`ere, where cover relations…
Planar coincidence site lattices and modules with N-fold symmetry are well understood in a formulation based on cyclotomic fields, in particular for the class number one case, where they appear as certain principal ideals in the…
The lattice path model suggested by E. Deutsch is derived from ordinary Dyck paths, but with additional down-steps of size -3,-5,-7,... . For such paths, we find the generating functions of them, according to length, ending at level $i$,…
We present an exact formula for the ordinary generating series of the simple paths between any two vertices of a graph. Our formula involves the adjacency matrix of the connected induced subgraphs and remains valid on weighted and directed…
We use a Hamiltonian (transition matrix) description of height-restricted Dyck paths on the plane in which generating functions for the paths arise as matrix elements of the propagator to evaluate the length and area generating function for…
The coincidence site lattices of the root lattice $A_4$ are considered, and the statistics of the corresponding coincidence rotations according to their indices is expressed in terms of a Dirichlet series generating function. This is…
Three-dimensional Catalan numbers are a variant of the classical (bidimensional) Catalan numbers, that count, among other interesting objects, the standard Young tableaux of shape (n,n,n). In this paper, we present a structural bijection…
We give a precise estimate for the number of lattice points in certain bounded subsets of $\mathbb{R}^{n}$ that involve `hyperbolic spikes' and occur naturally in multiplicative Diophantine approximation. We use Wilkie's o-minimal structure…
We introduce a bijection between inequivalent minimal factorizations of the n-cycle (1 2 ... n) into a product of smaller cycles of given length, on one side, and trees of a certain structure on the other. We use this bijection to count the…
We define a new lattice structure on the elements of a finite Coxeter group W. This lattice, called the shard intersection order, is weaker than the weak order and has the noncrossing partition lattice NC(W) as a sublattice. The new…
We study some distributive lattices arising in the combinatorics of lattice paths. In particular, for the Dyck, Motzkin and Schroder lattices we describe the spectrum and we determine explicitly the Euler characteristic in terms of natural…
Here we give two bijections, one to show that the number of UUU-free Dyck n-paths is the Motzkin number M_n, the other to obtain the (known) distributions of the parameters "number of UDUs" and "number of DDUs" on Dyck n-paths. The first…
We obtain the generating functions for partial matchings avoiding neighbor alignments and for partial matchings avoiding neighbor alignments and left nestings. We show that there is a bijection between partial matchings avoiding three…
We study the congruence lattices of the multinomial lattices L(v) introduced by Bennett and Birkhoff. Our main motivation is to investigate Parikh equivalence relations that model concurrent computation. We accomplish this goal by providing…