Advances in Applied Mathematics最新文献

英文中文

Extremal distance spectral radius of graphs with fixed size 固定尺寸图的极值距离谱半径

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-02-01 Epub Date: 2025-10-09 DOI: 10.1016/j.aam.2025.102980

Hongying Lin , Bo Zhou

Let m be a positive integer. Brualdi and Hoffman proposed the problem to determine the (connected) graphs with maximum adjacency spectral radius in a given graph class and they posed a conjecture for the class of graphs with given size m. After partial results due to Friedland and Stanley, Rowlinson completely confirmed the conjecture. The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. We investigate the problem to determine the connected graphs with minimum distance spectral radius in the class of graphs with size m. Given m, there is exactly one positive integer n such that

(\begin{matrix} n - 1 \\ 2 \end{matrix}) < m \leq (\begin{matrix} n \\ 2 \end{matrix})

. We establish some structural properties of the extremal graphs for all m and solve the problem for

(\begin{matrix} n - 1 \\ 2 \end{matrix}) + \max {\frac{n - 6}{2}, 1} \leq m \leq (\begin{matrix} n \\ 2 \end{matrix})

. We give a conjecture for the remaining case. To prove the main results, we also determine the complements of forests of fixed order with large and small distance spectral radius.

设m为正整数。Brualdi和Hoffman提出了确定给定图类中邻接谱半径最大的（连通）图的问题，并对给定大小为m的图类提出了一个猜想。在Friedland和Stanley的部分结果之后，Rowlinson完全证实了这个猜想。连通图的距离谱半径是其距离矩阵的最大特征值。我们研究了在大小为m的图类中确定具有最小距离谱半径的连通图的问题。给定m，存在一个正整数n使得（n−12）<m≤(n2)。我们建立了所有m的极值图的一些结构性质，并解决了(n−12)+max (n−62,1)≤m≤(n2)的问题。我们对剩下的情况作一个推测。为了证明主要结果，我们还确定了大小距离谱半径的定阶森林的补。

{"title":"Extremal distance spectral radius of graphs with fixed size","authors":"Hongying Lin , Bo Zhou","doi":"10.1016/j.aam.2025.102980","DOIUrl":"10.1016/j.aam.2025.102980","url":null,"abstract":"<div><div>Let m be a positive integer. Brualdi and Hoffman proposed the problem to determine the (connected) graphs with maximum adjacency spectral radius in a given graph class and they posed a conjecture for the class of graphs with given size m. After partial results due to Friedland and Stanley, Rowlinson completely confirmed the conjecture. The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. We investigate the problem to determine the connected graphs with minimum distance spectral radius in the class of graphs with size m. Given m, there is exactly one positive integer n such that <math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo><</mo><mi>m</mi><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math>. We establish some structural properties of the extremal graphs for all m and solve the problem for <math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>6</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>1</mn><mo>}</mo><mo>≤</mo><mi>m</mi><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math>. We give a conjecture for the remaining case. To prove the main results, we also determine the complements of forests of fixed order with large and small distance spectral radius.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102980"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268468","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The symmetric strong circuit elimination property 对称强电路消除特性

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-02-01 Epub Date: 2025-10-10 DOI: 10.1016/j.aam.2025.102983

Christine Cho , James Oxley , Suijie Wang

C_{1}

and

C_{2}

are circuits in a matroid M with

e_{1}

C_{1} - C_{2}

and e in

C_{1} \cap C_{2}

, then M has a circuit

C_{3}

such that

e \in C_{3} \subseteq (C_{1} \cup C_{2}) - e

. This strong circuit elimination axiom is inherently asymmetric. A matroid M has the symmetric strong circuit elimination property (SSCE) if, when the above conditions hold and

e_{2} \in C_{2} - C_{1}

, there is a circuit

C_{3}^{'}

with

{e_{1}, e_{2}} \subseteq C_{3}^{'} \subseteq (C_{1} \cup C_{2}) - e

. We prove that a connected matroid has this property if and only if it has no two skew circuits. We also characterize such matroids in terms of forbidden series minors, and we give a new matroid axiom system that is built around a modification of SSCE.

若C1和C2是矩阵M中的回路，其中e1在C1−C2中，e在C1∩C2中，则M存在一个回路C3，使得e∈C3蔓生(C1∪C2)−e。这个强电路消除公理本质上是不对称的。当满足上述条件，且e2∈C2−C1时，存在一个C3′≥{e1，e2}的 C3′≥(C1∪C2)−e的回路，则矩阵M具有对称强回路消去性（SSCE）。我们证明了一个连通的矩阵具有这个性质当且仅当它没有两个歪斜的电路。我们还用禁止级数的小调来描述这类拟阵，并给出了一个围绕SSCE的修正而建立的新的拟阵公理系统。

引用次数: 0

Hybrid pipe dreams for key polynomials 关键多项式的混合白日梦

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-02-01 Epub Date: 2025-10-08 DOI: 10.1016/j.aam.2025.102979

Yihan Xiao , Rui Xiong , Haofeng Zhang

We develop a family of new combinatorial models for key polynomials. It is similar to the hybrid pipe dream model for Schubert polynomials defined recently by Knutson and Udell.

我们开发了一组新的关键多项式组合模型。它类似于最近由Knutson和Udell定义的舒伯特多项式的混合白日梦模型。

引用次数: 0

d-Separated permutations and q-Stirling numbers of the first kind 第一类的d-分离排列和q-斯特林数

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-02-01 Epub Date: 2025-09-30 DOI: 10.1016/j.aam.2025.102976

Rosena R.X. Du, Yun Li

<div><div>Let d be a nonnegative integer, a d-separated permutation is a permutation in which every two left-to-right minima are at distance greater than d. More precisely, for <math><mi>π</mi><mo>=</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>π</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math>, suppose that <math><msub><mrow><mi>π</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>π</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>π</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub></math> are the left-to-right minima of π with <math><mi>k</mi><mo>≥</mo><mn>1</mn></math> and <math><mn>1</mn><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><mi>n</mi></math>, then π is d-separated if <math><msub><mrow><mi>i</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>−</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>></mo><mi>d</mi></math> for each j, <math><mn>1</mn><mo><</mo><mi>j</mi><mo>≤</mo><mi>k</mi></math>. In this paper we study different enumerative properties on d-separated permutations. We first give a recurrence formula of the numbers <math><msup><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math> of d-separated permutations in <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math> with exactly k left-to-right minima. Then we study the inversion and co-inversion polynomials of d-separated permutations, and give q-analogue <math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math> and <math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math>-analogue <math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math> of <math><msup><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math> for any d. Note that when <math><mi>d</mi><mo>=</mo><mn>0</mn></math>, 0-separated permutations are just all permutations in <math><msub><mrow><mi>S</mi></mrow

设d为非负整数，d分离置换是其中每两个从左到右极小值距离大于d的置换。更准确地说，对于π=π1π2⋯πn∈Sn，假设πi1，πi2，…，πik是π在k≥1且1=i1<i2<⋯<；ik≤n时的从左到右极小值，则对于每个j， 1<j≤k，如果ij−ij−1>；d，则π是d分离的。本文研究了d分隔排列的不同枚举性质。我们首先给出Sn中d分隔排列cd（n,k）的递推公式，从左到右的最小值正好为k。然后研究了d分离置换的反演多项式和共反演多项式，给出了任意d下cd（n,k）的q-analogue cqd（n,k）和(p,q)-analogue cp,qd（n,k）。注意，当d=0时，0-separated置换只是Sn中的所有置换，c0（n,k）是第一类Stirling数，两个多项式cqd（n,k）和cp，1d（n,k）都是第一类q-Stirling数的推广。当d=0时，第一类q-Stirling数已经被m -迪斯-勒鲁和凯-雷迪很好地研究过，他们通过在阶梯棋盘上放置车给出了很好的组合解释。我们给出了阶梯棋盘中排列和车的位置之间的关系。

引用次数: 0

A class of generalized chord Minkowski problems 一类广义弦Minkowski问题

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-02-01 Epub Date: 2025-10-06 DOI: 10.1016/j.aam.2025.102978

Lu Zhang

In this paper, we consider a class of generalized chord integrals in integral geometry, where the integrand is a generalized kernel that replaces the Riesz kernel. The generalized chord measure arises from the study of the generalized chord integral of convex bodies. We pose the Minkowski problem for the generalized chord measure and obtain the existence of solutions to the related Minkowski problem.

本文研究了积分几何中的一类广义弦积分，其中被积函数是一个广义核，它代替了Riesz核。广义弦测度来源于对凸体广义弦积分的研究。提出了广义弦测度的Minkowski问题，得到了相关Minkowski问题解的存在性。

引用次数: 0

Degeneration in discriminantal arrangements 歧视性安排的退化

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-02-01 Epub Date: 2025-11-24 DOI: 10.1016/j.aam.2025.103001

Takuya Saito

Discriminantal arrangements are hyperplane arrangements that are generalization of braid arrangements. They are constructed from given hyperplane arrangements, but their combinatorics are not invariant under combinatorial equivalence. However, it is known that the combinatorics of the discriminantal arrangements are constant on a Zariski open set of the space of hyperplane arrangements. In the present paper, we introduce

(T, r)

-singularity varieties in the space of hyperplane arrangements to classify discriminantal arrangements and show that the Zariski open set is the complement of

(T, r)

-singularity varieties. We study their basic properties and operations and provide examples, including infinite families of

(T, r)

-singularity varieties. In particular, the operation that we call degeneration is a powerful tool for constructing

(T, r)

-singularity varieties. As an application, we provide a list of

(T, r)

-singularity varieties for spaces of small line arrangements.

判别排列是超平面排列，是辫状排列的推广。它们是由给定的超平面排列构造而成，但在组合等价条件下，它们的组合不是不变的。然而，在超平面排列空间的Zariski开集上，判别排列的组合是常数。本文在超平面排列空间中引入(T,r)-奇异变异体来对判别式排列进行分类，并证明Zariski开集是(T,r)-奇异变异体的补集。我们研究了它们的基本性质和运算，并举例说明了(T,r)-奇点的无穷族。特别地，我们称之为退化的操作是构造(T,r)-奇点变体的有力工具。作为一种应用，我们给出了小线排列空间的(T,r)-奇点变体的列表。

{"title":"Degeneration in discriminantal arrangements","authors":"Takuya Saito","doi":"10.1016/j.aam.2025.103001","DOIUrl":"10.1016/j.aam.2025.103001","url":null,"abstract":"<div><div>Discriminantal arrangements are hyperplane arrangements that are generalization of braid arrangements. They are constructed from given hyperplane arrangements, but their combinatorics are not invariant under combinatorial equivalence. However, it is known that the combinatorics of the discriminantal arrangements are constant on a Zariski open set of the space of hyperplane arrangements. In the present paper, we introduce <math><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math>-singularity varieties in the space of hyperplane arrangements to classify discriminantal arrangements and show that the Zariski open set is the complement of <math><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math>-singularity varieties. We study their basic properties and operations and provide examples, including infinite families of <math><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math>-singularity varieties. In particular, the operation that we call degeneration is a powerful tool for constructing <math><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math>-singularity varieties. As an application, we provide a list of <math><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math>-singularity varieties for spaces of small line arrangements.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 103001"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A Brualdi-Hoffman-Turán problem on theta graph 图上的Brualdi-Hoffman-Turán问题

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-02-01 Epub Date: 2025-11-18 DOI: 10.1016/j.aam.2025.103000

Chang Liu , Jianping Li , Shuchao Li , Yuantian Yu

<div><div>The Brualdi-Hoffman-Turán problem, a central topic in spectral graph theory, seeks to determine maximum spectral radius <math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> of an F-free graph G with m edges. This problem has attracted significant attention in recent years. Let <math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math> denote the theta graph obtained by adding a chord between two vertices at distance 2 on cycle <math><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow></msub></math>. Zhai, Lin, and Shu [32] conjectured that, for <math><mi>k</mi><mo>⩾</mo><mn>2</mn></math> and sufficiently large m, if G is <math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math>-free or <math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></msub></math>-free, then <math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>⩽</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>m</mi><mo>−</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></math>, with equality if and only if <math><mi>G</mi><mo>≅</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∨</mo><mo>(</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math>. This conjecture was highlighted in Liu and Ning's survey [18] as one of the twenty unsolved problems in spectral graph theory. Subsequently, Y.T. Li proposed an even stronger conjecture, which claims that the upper bound on <math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> and the corresponding extremal graph hold for <math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math>-free, or <math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math>-free graphs. Recently, Li, Zhai, and Shu [14] resolved both conjectures completely. Note that the above extremal graph <math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∨</mo><mo>(</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math> is well-defined only if <math><mi>m</mi><mo>+</mo><mfrac><mrow><mi>k</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</

Brualdi-Hoffman-Turán问题是谱图理论中的一个中心问题，它试图确定具有m条边的无f图G的最大谱半径ρ(G)。这个问题近年来引起了极大的关注。设Cl+表示在圆Cl上距离为2的两个顶点之间加上弦得到的图。Zhai， Lin和Shu[32]推测，对于k小于或等于2且足够大的m，如果G是C2k+1-free或C2k+2-free，则ρ(G)≤k−1+4m−k2+12，当且仅当G≠Kk∨(mk−k−12)K1时相等。这一猜想在刘和宁的调查[18]中被列为谱图理论中未解决的二十个问题之一。随后，Y.T. Li提出了一个更强的猜想，即对于C2k+1+ free，或C2k+2+ free图，ρ(G)的上界和相应的极值图都成立。最近，Li， Zhai和Shu b[14]完全解决了这两个猜想。注意上述极值图Kk∨(mk−k−12)K1只有在m+k(k+1)2≡0（modk）时是定义良好的。因此，考虑以下问题是很自然的：在m+k(k+1)2≡l（modk）有1≤l<；k的条件下，G是一个有m条边的C2k+1+自由或C2k+2+自由的图，我们能否确定ρ(G)的尖锐上界？本文采用k核法和光谱技术解决了这一问题。我们的结果推广了上述两个猜想。

引用次数: 0

Knuth's big-chooser matchbox process: the case of many matchboxes Knuth的大选择火柴盒过程：许多火柴盒的情况

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-02-01 Epub Date: 2025-11-06 DOI: 10.1016/j.aam.2025.102996

Mark Dukes, Andrew Mullins

Banach's matchbox problem considers the setting of two matchboxes that each initially contain the same number of matches. Boxes are chosen with equal probability and a match removed each time. The problem concerns the law of the number of matches remaining in one box once the other box empties. Knuth considered a generalization of this problem whereby big-choosers arrive with probability p and remove a match from the box with the most number remaining, and little-choosers arrive with probability

1 - p

and remove a match from the box with the least number remaining.

In this paper we consider Knuth's generalization for the case of k matchboxes in which there are big-choosers and little-choosers. We determine the generating function for the expected number of matches remaining in

k - 1

matchboxes once a box first empties, a quantity we refer to as the ‘residue’. Interestingly, this generating function is a quotient whose denominator contains a generating function for a special case of the Raney numbers. The form for this generating function allows us to give an expression for the expected residue in terms of a sum that involves diagonal state return probabilities, where a diagonal state is a configuration in which all matchboxes each contain the same number of matches. We use analytic techniques to determine the asymptotic behavior of this expected value for all values of p, which involves the study of an asymmetric random walk.

In addition to this we consider the expected value of the order of the first return to a diagonal state and determine the asymptotic behavior of this quantity. The coefficients of the diagonal state probability generating function are shown to be related to ‘manila folder configurations in a filing cabinet’, and we make this connection precise. This allows us to use known results for the enumeration of such manila folder configurations to give a closed form expression for the diagonal state return probabilities.

Banach的火柴盒问题考虑了两个火柴盒的设置，每个火柴盒最初都包含相同数量的火柴。选择盒子的概率相等，每次取出一根火柴。这个问题涉及到一个盒子里剩下的火柴数的规律，当另一个盒子空了。Knuth考虑了这个问题的一个推广，即大选择者以p的概率到达并从剩余数量最多的盒子中取出一根火柴，而小选择者以1−p的概率到达并从剩余数量最少的盒子中取出一根火柴。本文考虑了k个火柴盒存在大挑挑者和小挑挑者的情况下Knuth的推广。当一个火柴盒第一次清空时，我们确定k−1个火柴盒中剩余火柴的预期数量的生成函数，我们将这个数量称为“剩余”。有趣的是，这个生成函数是一个商，它的分母包含一个用于兰尼数特殊情况的生成函数。这个生成函数的形式允许我们以对角状态返回概率的和的形式给出期望剩余的表达式，其中对角状态是所有火柴盒中每个火柴盒包含相同数量的火柴的配置。我们使用解析技术来确定该期望值对所有p值的渐近行为，这涉及到不对称随机漫步的研究。除此之外，我们还考虑了第一次返回对角线状态的阶数的期望值，并确定了该量的渐近行为。对角线状态概率生成函数的系数显示与“文件柜中的马尼拉文件夹配置”相关，并且我们使这种连接精确。这允许我们使用已知的结果来枚举这样的马尼拉文件夹配置，从而给出对角线状态返回概率的封闭形式表达式。

{"title":"Knuth's big-chooser matchbox process: the case of many matchboxes","authors":"Mark Dukes, Andrew Mullins","doi":"10.1016/j.aam.2025.102996","DOIUrl":"10.1016/j.aam.2025.102996","url":null,"abstract":"<div><div>Banach's matchbox problem considers the setting of two matchboxes that each initially contain the same number of matches. Boxes are chosen with equal probability and a match removed each time. The problem concerns the law of the number of matches remaining in one box once the other box empties. Knuth considered a generalization of this problem whereby big-choosers arrive with probability p and remove a match from the box with the most number remaining, and little-choosers arrive with probability <math><mn>1</mn><mo>−</mo><mi>p</mi></math> and remove a match from the box with the least number remaining.</div><div>In this paper we consider Knuth's generalization for the case of k matchboxes in which there are big-choosers and little-choosers. We determine the generating function for the expected number of matches remaining in <math><mi>k</mi><mo>−</mo><mn>1</mn></math> matchboxes once a box first empties, a quantity we refer to as the ‘residue’. Interestingly, this generating function is a quotient whose denominator contains a generating function for a special case of the Raney numbers. The form for this generating function allows us to give an expression for the expected residue in terms of a sum that involves diagonal state return probabilities, where a diagonal state is a configuration in which all matchboxes each contain the same number of matches. We use analytic techniques to determine the asymptotic behavior of this expected value for all values of p, which involves the study of an asymmetric random walk.</div><div>In addition to this we consider the expected value of the order of the first return to a diagonal state and determine the asymptotic behavior of this quantity. The coefficients of the diagonal state probability generating function are shown to be related to ‘manila folder configurations in a filing cabinet’, and we make this connection precise. This allows us to use known results for the enumeration of such manila folder configurations to give a closed form expression for the diagonal state return probabilities.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102996"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145474010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The optimal upper bound on the MP-ratio for quaternary words 四元词的MP-ratio的最优上界

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-02-01 Epub Date: 2025-10-21 DOI: 10.1016/j.aam.2025.102984

Kristina Ago, Bojan Bašić

The so-called MP-ratio is a kind of measure of how “packed with palindromes” a given word is. The lower bound on the MP-ratio for the set of all n-ary words is (trivially) 1, while the best possible upper bound is an open problem in the general case. It is solved for

n = 2

(where the optimal upper bound is 4) and for

n = 3

(where the optimal upper bound is 6). Also, it is known that in the n-ary case the optimal bound is between 2n and the order of growth

n 2^{\frac{n}{2}}

. In this article we solve this problem for quaternary words, for which we show that the best possible upper bound on the MP-ratio equals 8. We believe that this is the last case in which the result is 2n, that is, we believe that for

n ⩾ 5

there are words whose MP-ratio is strictly larger than 2n.

所谓的mp比率是一种衡量给定单词“回文堆积”程度的方法。所有n元词集合的MP-ratio的下界（通常）是1，而在一般情况下，最佳可能上界是一个开放问题。对于n=2（其中最优上界是4）和n=3（其中最优上界是6），可以求解。此外，已知在n元情况下，最优边界在2n和增长阶数n2n2之间。在本文中，我们解决了四元词的这个问题，我们证明了MP-ratio的最佳可能上界等于8。我们相信这是最后一个结果为2n的案例，也就是说，我们相信对于n大于等于5的单词，其mp比率严格大于2n。

引用次数: 0

Lattice minors and Eulerian posets 格子集和欧拉偏集

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-02-01 Epub Date: 2025-11-15 DOI: 10.1016/j.aam.2025.102997

William Gustafson

We introduce posets of simple vertex labeled minors of graphs and a generalization to the level of polymatroids, collectively termed minor posets. We show that any minor poset is isomorphic to the face poset of a regular CW sphere, and in particular, is Eulerian. We establish cd-index inequalities induced by strong maps, a tight upper bound for cd-indices of minor posets and a tight lower bound for cd-indices of minor posets arising from lattices of maximal length.

我们引入了标记为图的子点的简单顶点的偏集，并将其推广到多拟阵的水平，统称为子偏集。我们证明了任意小偏序与正则连续波球的面偏序是同构的，特别是它是欧拉的。我们建立了由强映射引起的cd-指标不等式，以及由极大长度格产生的小偏序集的cd-指标的紧上界和小偏序集的cd-指标的紧下界。

引用次数: 0

首页上一页

下一页尾页

类型

全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

数据库

全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

期刊

Advances in Applied Mathematics

全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.

﹀