Promotion of Kreweras words

Kreweras words are words consisting of n$$\mathrm {A}$$ A ’s, n$$\mathrm {B}$$ B ’s, and n$$\mathrm {C}$$ C ’s in which every prefix has at least as many $$\mathrm {A}$$ A ’s as $$\mathrm {B}$$ B ’s and at least as many $$\mathrm {A}$$ A ’s as $$\mathrm {C}$$ C ’s. Equivalently, a Kreweras word is a linear extension of the poset $$\mathsf{V}\times [n]$$ V × [ n ] . Kreweras words were introduced in 1965 by Kreweras, who gave a remarkable product formula for their enumeration. Subsequently they became a fundamental example in the theory of lattice walks in the quarter plane. We study Schützenberger’s promotion operator on the set of Kreweras words. In particular, we show that 3n applications of promotion on a Kreweras word merely swaps the $$\mathrm {B}$$ B ’s and $$\mathrm {C}$$ C ’s. Doing so, we provide the first answer to a question of Stanley from 2009, asking for posets with ‘good’ behavior under promotion, other than the four families of shapes classified by Haiman in 1992. We also uncover a strikingly simple description of Kreweras words in terms of Kuperberg’s $$\mathfrak {sl}_3$$ sl 3 -webs, and Postnikov’s trip permutation associated with any plabic graph. In this description, Schützenberger’s promotion corresponds to rotation of the web.

Mating of Discrete Trees and Walks in the Quarter-Plane

We give a general construction of triangulations starting from a walk in the quarter plane with small steps, which is a discrete version of the mating of trees. We use a special instance of this construction to give a bijection between maps equipped with a rooted spanning tree and walks in the quarter plane. We also show how the construction allows to recover several known bijections between such objects in a uniform way.

A New Approach to Preform Design in Metal Forging Processes Based on the Convolution Neural Network

This study presents an innovative methodology for preform design in metal forging processes based on the convolution neural network (CNN) algorithm. The proposed approach extracts the features of inputted forging product geometries and utilizes them to derive the corresponding preform shapes by employing weight arrays (filters) determined during the convolutional operations. The filters are progressively updated during the training process, emulating the learning steps of a process engineer responsible for the design of preform shapes for the forging processes. The design system is composed of multiple three-dimensional (3D) CNN sub-models, which can automatically derive individual 3D preform design candidates. It also implies that the 3D surfaces of preforms are easily acquired, which is important for the forging industry. The proposed preform design methodology was validated by applying it to two-dimensional (2D) axisymmetric shapes, one-quarter plane-symmetric 3D shapes, and two other industrial cases. In all the considered cases, the design methodology achieved substantial reductions in the forging load without forging defects, proving its reliability and effectiveness for application in metal forging processes.

On Green’s function of Cauchy–Dirichlet problem for hyperbolic equation in a quarter plane

The definition of a Green’s function of a Cauchy–Dirichlet problem for the hyperbolic equation in a quarter plane is given. Its existence and uniqueness have been proven. Representation of the Green’s function is given. It is shown that the Green’s function can be represented by the Riemann–Green function.