H-admissible permutations and the HCP

Admissible pinnacle orderings

A pinnacle of a permutation is a value that is larger than its immediate neighbors when written in one-line notation. In this paper, we build on previous work that characterized admissible pinnacle sets of permutations. For these sets, there can be specific orderings of the pinnacles that are not admissible, meaning that they are not realized by any permutation. Here we characterize admissible orderings, using the relationship between a pinnacle x and its rank in the pinnacle set to bound the number of times that the pinnacles less than or equal to x can be interrupted by larger values.

Pinnacle sets of signed permutations

Pinnacle sets record the values of the local maxima for a given family of permutations. They were introduced by Davis-Nelson-Petersen-Tenner as a dual concept to that of peaks, previously defined by Billey-Burdzy-Sagan. In recent years pinnacles and admissible pinnacles sets for the type $A$ symmetric group have been widely studied. In this article we define the pinnacle set of signed permutations of types $B$ and $D$. We give a closed formula for the number of type $B$/$D$ admissible pinnacle sets and answer several other related enumerative questions.

Change Ringing and Hamiltonian Cycles : The Search for Erin and Stedman Triples

A very old problem in campanology is the search for peals. The latter can be thought of as a heavily constrained sequence of all possible permutations of a given size, where the exact nature of the constraints depends on which method of ringing is desired. In particular, we consider the methods of bobs-only Stedman Triples and Erin Triples; the existence of the latter is still an open problem. We show that this problem can be viewed as a similarly constrained form of the Hamiltonian cycle problem (HCP). Through the use of special subgraphs, we convert this to a standard instance of HCP. The or…

Erdős-Graham两集合置换问题中的结构刚性

Structural rigidity in the Erdős-Graham two-set permutation problem

本文针对Erdős-Graham两集合置换问题，通过研究二元候选集S_A的结构限制，证明其不存在可允许置换，表明典型二元划分无法解决该问题，该问题仍待解决。

Convex cones and SAGBI bases of permutation invariants

Let G be a permutation group acting on {1,...,n}, and < be any admissible term order on the polynomial ring K[x_1,...,x_n]. We prove that the invariant ring K[x_1,...,x_n]^G of G has a finite SAGBI basis if, and only if, G is generated by reflections.

棒棒糖图相关的 Hessenberg 簇上同调的置换模分解

Permutation module decomposition of the cohomology of Hessenberg varieties associated with lollipop graphs

研究棒棒糖图相关正则半单 Hessenberg 簇上同调作为点作用下的模，利用 CHL 基建立点作用结构性质，得到棒棒糖图色拟对称函数展开，构建置换模分解，证明相关猜想。