四类associahedra模型之间的等价性

A generalised Euler-Poincaré formula for associahedra

We derive a formula for the number of flip-equivalence classes of tilings of an $n$-gon by collections of tiles of shape dictated by an integer partition $λ$. The proof uses the Euler-Poincaré formula; and the formula itself generalises the Euler-Poincaré formula for associahedra.

Coloring Planar Graphs via Colored Paths in the Associahedra

Hassler Whitney's theorem of 1931 reduces the task of finding proper, vertex 4-colorings of triangulations of the 2-sphere to finding such colorings for the class \(\mathfrak H\) of triangulations of the 2-sphere that have a Hamiltonian circuit. This has been used by Whitney and others from 1936 to the present to find equivalent reformulations of the 4 Color Theorem (4CT). Recently there has been activity to try to use some of these reformuations to find a shorter proof of the 4CT. Every triangulation in \(\mathfrak H\) has a dual graph that is a union of two binary trees with the same number …

Generalizing the Splits Equivalence Theorem and Four Gamete Condition: Perfect Phylogeny on Three State Characters

We study the perfect phylogeny problem and establish a generalization of the four gamete condition (also called the Splits Equivalence Theorem) for sequences over three state characters. Our main result is that a set of input sequences over three state characters allows a perfect phylogeny if and only if every subset of three characters allows a perfect phylogeny. In establishing these results, we prove fundamental structural features of the perfect phylogeny problem on three state characters and completely characterize the minimal obstruction sets that must occur in three state input sequence…

排列模式中的若干问题：单峰性、等价性、错排和排序

An assortment of problems in permutation patterns: unimodality, equivalence, derangements, and sorting

本文探讨了排列模式中的单峰性、等价性、错排计数及排序问题，提出四个开放研究方向。

Supplement to "Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs"

This supplementary material includes three parts: some preliminary results, four examples, an experiment, three new algorithms, and all proofs of the results in the paper "Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs".

Complementary Algorithms For Tableaux

We study four operations defined on pairs of tableaux. Algorithms for the first three involve the familiar procedures of jeu de taquin, row insertion, and column insertion. The fourth operation, hopscotch, is new, although specialised versions have appeared previously. Like the other three operations, this new operation may be computed with a set of local rules in a growth diagram, and it preserves Knuth equivalence class. Each of these four operations gives rise to an a priori distinct theory of dual equivalence. We show that these four theories coincide. The four operations are linked via th…