Annals of Combinatorics最新文献_第6页

Completing the Asymptotic Classification of Mostly Symmetric Short Step Walks in an Orthant 完成正交上大部分对称短步行走的渐近分类

IF 0.7 4区数学

Annals of Combinatorics Pub Date : 2024-12-20 DOI: 10.1007/s00026-024-00739-6

Alexander Kroitor, Stephen Melczer

引用次数: 0

From Hodge Theory for Tame Functions to Ehrhart Theory for Polytopes 从驯服函数的Hodge理论到多面体的Ehrhart理论

IF 0.7 4区数学

Annals of Combinatorics Pub Date : 2024-12-12 DOI: 10.1007/s00026-024-00738-7

Antoine Douai

{"title":"From Hodge Theory for Tame Functions to Ehrhart Theory for Polytopes","authors":"Antoine Douai","doi":"10.1007/s00026-024-00738-7","DOIUrl":"10.1007/s00026-024-00738-7","url":null,"abstract":"<div>We study the interplay between Sabbah’s mixed Hodge structure for tame regular functions and Ehrhart theory for polytopes. We first analyze the Poincaré polynomial of the Hodge filtration of this mixed Hodge structure (we call this Poincaré polynomial the (theta )-vector). Using the symmetry of the Hodge numbers involved, we show that it shares many properties with the (h^*)-vector of a polytope. For instance, we define from the (theta )-vector the Hodge–Ehrhart polynomial of a general tame function and we show that it satisfies a reciprocity law, analogous to the one satisfied by the Ehrhart polynomial of a polytope. We study the roots of this Hodge–Ehrhart polynomial, in particular their distribution around some critical lines. Using techniques coming from singularity theory, we also show a Thom–Sebastiani type theorem for the (theta )-vector. Finally, we offer some linear inequalities among the coefficients of the (theta )-vectors which could be helpful to test if a polynomial is a (theta )-vector or not. In the very particular case of convenient and nondegenerate Laurent polynomials, we show (using the Brieskorn lattice and the V-filtration) that the previous results agree with the classical ones in combinatorics and we emphasize various combinatorial properties of Sabbah’s Hodge numbers: on the way, this provides an alternative interpretation of prior results about the (limit) Hodge numbers of hypersurfaces in a torus obtained in a different framework by Danilov–Khovanskiĭ and more recently by Katz–Stapledon.</div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"29 4","pages":"1077 - 1107"},"PeriodicalIF":0.7,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145435846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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

Local Limit Theorems for Hook Lengths in Partitions 分区中钩子长度的局部极限定理

IF 0.7 4区数学

Annals of Combinatorics Pub Date : 2024-12-11 DOI: 10.1007/s00026-024-00737-8

Tapas Bhowmik, Wei-Lun Tsai

{"title":"Local Limit Theorems for Hook Lengths in Partitions","authors":"Tapas Bhowmik, Wei-Lun Tsai","doi":"10.1007/s00026-024-00737-8","DOIUrl":"10.1007/s00026-024-00737-8","url":null,"abstract":"<div>Partition hook lengths have wide-ranging applications in combinatorics, number theory, physics, and representation theory. We study two infinite families of random variables associated with t-hooks. For fixed (tge 1,) if (Y_{t;,n}) counts the number of hooks of length t in a random integer partition of n, we prove a uniform local limit theorem for (Y_{t;,n}) on any bounded set of ({mathbb {R}}.) To achieve this, we establish an asymptotic formula with a power-saving error term for the number of partitions of n with m many t-hooks. In contrast, we define ({widehat{Y}}_{t;,n}) as the count of hooks divisible by t in a randomly chosen partition of n. While ({widehat{Y}}_{t;,n}) converges in distribution, we show that it fails to satisfy the local limit theorem for any (t ge 2). The proofs employ the multivariable saddle-point method, asymptotic formulas for the number of t-core partitions from Anderson and Lulov–Pittel, and estimates of certain exponential sums. Notably, for (t=4,) the analysis involves the asymptotic behavior of class numbers of imaginary quadratic fields.</div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"29 3","pages":"853 - 884"},"PeriodicalIF":0.7,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-024-00737-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145037274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Special Functions for Hyperoctahedral Groups Using Bosonic Lattice Models 基于玻色子晶格模型的超八面体群的特殊函数

IF 0.7 4区数学

Annals of Combinatorics Pub Date : 2024-12-09 DOI: 10.1007/s00026-024-00734-x

Ben Brubaker, Will Grodzicki, Andrew Schultz

引用次数: 0

On Partitions with Bounded Largest Part and Fixed Integral GBG-rank Modulo Primes 关于最大部有界的分区和固定积分gbg -秩模素数

IF 0.7 4区数学

Annals of Combinatorics Pub Date : 2024-12-09 DOI: 10.1007/s00026-024-00733-y

Alexander Berkovich, Aritram Dhar

引用次数: 0

Words Avoiding Tangrams 避免拼字

IF 0.7 4区数学

Annals of Combinatorics Pub Date : 2024-12-05 DOI: 10.1007/s00026-024-00736-9

Michał Dȩbski, Jarosław Grytczuk, Bartłomiej Pawlik, Jakub Przybyło, Małgorzata Śleszyńska-Nowak

{"title":"Words Avoiding Tangrams","authors":"Michał Dȩbski, Jarosław Grytczuk, Bartłomiej Pawlik, Jakub Przybyło, Małgorzata Śleszyńska-Nowak","doi":"10.1007/s00026-024-00736-9","DOIUrl":"10.1007/s00026-024-00736-9","url":null,"abstract":"<div>A tangram is a word in which every letter occurs an even number of times. Such word can be cut into parts that can be arranged into two identical words. The minimum number of cuts needed is called the cut number of a tangram. For example, the word <img> is a tangram with cut number one, while the word <img> is a tangram with cut number two. Clearly, tangrams with cut number one coincide with the well-known family of words, known as squares, having the form UU for some nonempty word U. A word W avoids a word T if it is not possible to write (W=ATB), for any words A and B (possibly empty). The famous 1906 theorem of Thue asserts that there exist arbitrarily long words avoiding squares over an alphabet with just three letters. Given a fixed number (kgeqslant 1), how many letters are needed to avoid tangrams with the cut number at most k? Let t(k) denote the minimum size of an alphabet needed for that purpose. By Thue’s result we have (t(1)=3), which easily implies (t(2)=3). Curiously, these are currently the only known exact values of this function. In our main result we prove that (t(k)=Theta (log _2k)). The proof uses entropy compression argument and Zimin words. Using a different method we prove that (t(k)leqslant k+1) for all (kgeqslant 4), which gives more exact estimates for small values of k. The proof makes use of Dejean words and a curious property of Gauss words, which is perhaps of independent interest.</div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"29 3","pages":"905 - 920"},"PeriodicalIF":0.7,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-024-00736-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145037268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Planar Binary Trees, Noncrossing Partitions and the Operator-Valued S-Transform 平面二叉树、非交叉划分和算子值s变换

IF 0.7 4区数学

Annals of Combinatorics Pub Date : 2024-11-28 DOI: 10.1007/s00026-024-00730-1

Kurusch Ebrahimi-Fard, Timothé Ringeard

引用次数: 0

Two-Face Complete Regular Dessins 双面完成常规设计

IF 0.7 4区数学

Annals of Combinatorics Pub Date : 2024-11-24 DOI: 10.1007/s00026-024-00735-w

Yuanyuan Chen, Wenwen Fan, Li Ma

引用次数: 0

General Position Sets, Colinear Sets, and Sierpiński Product Graphs 一般位置集，共线性集，和Sierpiński产品图