Plane partitions in the work of Richard Stanley and his school
英文摘要:
These notes provide a survey of the theory of plane partitions, seen through the glasses of the work of Richard Stanley and his school.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
C. Krattenthaler
These notes provide a survey of the theory of plane partitions, seen through the glasses of the work of Richard Stanley and his school.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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I will share with the reader what I have learned from Richard Stanley and the ways in which he has contributed to research in combinatorics conducted by me and my collaborators.
This short tribute to the guru of Enumerative and Algebraic Combinatorics started out when one the authors(DZ) attended the Stanely@70 conference, that took place at the same time as the preliminary stage of the 2014 World Cup. It states a surprising application of an analog of Richard Stanley's famous theorem about the enumeration of magic squares to the enumeration of possible outcomes in a World Cup Group.
We review Stanley's seminal work on the number of reduced words of the longest element of the symmetric group and his Stanley symmetric functions. We shed new light on this by giving a crystal theoretic interpretation in terms of decreasing factorizations of permutations. Whereas crystal operators on tableaux are coplactic operators, the crystal operators on decreasing factorization intertwine with the Edelman-Greene insertion. We also view this from a random perspective and study a Markov chain on reduced words of the longest element in a finite Coxeter group, in particular the symmetric grou…
Richard Stanely proposed, in a recent Amer. Math. Monthly Problem, to prove a nice explicit formula for the generating function for the number of n-letter words in {H,T} that have as many occurrences of HT as HH. In this article, we show how to prove this problem automatically, and ANY problem of that type, regardless of the size of the alphabet and the length of the two chosen strings
In a delightful article, Richard Stanley derived, algebraically, the surprisingly simple formula, 3 times 7 to the power n-1, for the sum of the cubes of the n-th row of Stern's diatomic array. In this note, we find an elegant bijective proof of this surprising fact, that explains it and gives insight. The novelty is that this gorgeous bijection was discovered by a computer (SBE), with minimal guidance by a human (DZ). This debunks the conventional wisdom, held by some human supremacists, that computers can only compute, but they can't give insight
In How Ways Can You Play Stanley Solitaire?
本文介绍了纪念Richard Stanley的简单单人纸牌游戏Stanley Solitaire,证明了其玩法数量的显式闭式公式,现有证明依赖Stanley 1984年的深刻定理,并邀读者寻找更初等的证明。
Richard Stanley played a crucial role, through his work and his students, in the development of the relatively new area known as combinatorial representation theory. In the early stages, he has the merit to have pointed out to combinatorialists the potential that representation theory has for applications of combinatorial methods. Throughout his distinguished career, he wrote significant articles which touch upon various combinatorial aspects related to representation theory (of Lie algebras, the symmetric group, etc.). I describe some of Richard's contributions involving Lie algebras, as well…
We give two simple proofs of a conjecture of Richard Stanley concerning the equidistribution of derangements and alternating permutations with the maximal number of fixed points.
Richard Stanley proved that the centralized/normalized version of the random variable "length of largest up-down subsequence" in a random permutation of length n is asymptotically normal. We go beyond and present a more refined asymptotic formula for the moments.
Automated Generation of Generating Functions Related to Generalized Stern's Diatomic Arrays in the footsteps of Richard Stanley
本文提出并实现了基于符号动态规划的 Maple 算法,用于自动生成广义 Stern 阵列的生成函数,并将其推广到由常系数线性递推序列定义的更一般阵列。
In his paper, "On a Partition Function of Richard Stanley," George Andrews proves a certain partition identity analytically and asks for a combinatorial proof. This paper provides the requested combinatorial proof.
This expository paper features a few highlights of Richard Stanley's extensive work in Ehrhart theory, the study of integer-point enumeration in rational polyhedra. We include results from the recent literature building on Stanley's work, as well as several open problems.
The Stanley Conjecture Revisited
本文基于 Stanley 猜想,提出 Jack Littlewood-Richardson 系数的窗口化性质,并将 Schur L-R 的分解性质推广到 Jack 情形,从而建立大量关系并实现一类情形的直接计算。
We give a historical survey of the theory P-partitions, starting with MacMahon's work, describing Richard Stanley's contributions and his related work, and continuing with more recent developments.
Let the sign of a standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. A conjecture by Richard Stanley says that the sum of the signs of all SYTs with n squares is 2^[n/2]. We present a stronger theorem with a purely combinatorial proof using the Robinson-Schensted correspondence and a new concept called chess tableaux. We also prove a sharpening of another conjecture by Stanley concerning weighted sums of squares of sign-imbalances. The proof is built on a remarkably simple relation between the sign of a permutation and the si…
Richard P. Stanley conjectured that finite trees can be distinguished by their chromatic symmetric functions. In this paper, we prove an analogous statement for posets: Finite rooted trees can be distinguished by their order quasisymmetric functions.
New cases of the Strong Stanley Conjecture
本文研究 Jack 对称函数的 Littlewood-Richardson 系数,证明矩形并情形下的 Strong Stanley 猜想,给出上下钩长乘积公式,并猜想其对 Shifted Jack 函数成立。
In the historic conference Combinatoire Enumerative[LL] wonderfully organized by Gilbert Labelle and Pierre Leroux there were many stimulating lectures, including a very interesting one by Pierre Leroux himself, who talked about his joint work with Xavier Viennot[LV], on solving differential equations combinatorially! During the problem session of that very same colloque, chaired by Pierre Leroux, Richard Stanley raised some intriguing problems about the enumeration of plane partitions, that he later expanded into a fascinating article[Sta1]. Most of these problems concerned the enumeration of…
Recently Richard Stanley initiated a study of the distribution of the length as(w) of the longest alternating subsequence in a random permutation w from the symmetric group $S_n$. Among other things he found an explicit formula for the generating function (on n and k) for the probability that as(w) is at most k and conjectured that the distribution, suitably centered and normalized, tended to a Gaussian with variance 8/45. In this note we present a proof of the conjecture based on the generating function.
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