Robert Davis, Akihiro Higashitani, Hidefumi Ohsugi
{"title":"On the Ehrhart Theory of Generalized Symmetric Edge Polytopes","authors":"Robert Davis, Akihiro Higashitani, Hidefumi Ohsugi","doi":"10.1007/s00026-026-00835-9","DOIUrl":"10.1007/s00026-026-00835-9","url":null,"abstract":"<div><p>The symmetric edge polytope (SEP) of a (finite, undirected) graph is a centrally symmetric lattice polytope whose vertices are defined by the edges of the graph. SEPs have been studied extensively in the past twenty years. Recently, Tóthmérész and, independently, D’Alí, Juhnke-Kubitzke, and Koch generalized the definition of an SEP to regular matroids, which are the matroids that can be represented by totally unimodular matrices. Generalized SEPs are known to have symmetric Ehrhart <span>(h^*)</span>-polynomials, and Ohsugi and Tsuchiya conjectured that (ordinary) SEPs have nonnegative <span>(gamma )</span>-vectors. In this article, we use combinatorial and Gröbner basis techniques to extend additional known properties of SEPs to generalized SEPs. Along the way, we show that generalized SEPs are not necessarily <span>(gamma )</span>-nonnegative by providing explicit examples. We prove that the polytopes we construct are “nearly” <span>(gamma )</span>-nonnegative in the sense that, by deleting two particular elements from the matroid, one obtains SEPs for graphs that are <span>(gamma )</span>-nonnegative. This provides further evidence that Ohsugi and Tsuchiya’s conjecture holds in the ordinary case.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"30 3","pages":"601 - 634"},"PeriodicalIF":0.8,"publicationDate":"2026-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149007370","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}
{"title":"Tree-Robinson Dissimilarities","authors":"Pascal Préa","doi":"10.1007/s00026-026-00829-7","DOIUrl":"10.1007/s00026-026-00829-7","url":null,"abstract":"<div><p>Given a dissimilarity <span>(textbf{d})</span> on a <i>n</i>-set <i>X</i>, a tree <i>T</i> with vertex set <i>X</i> is said to be <i>R-compatible</i> with <span>((X, textbf{d}))</span> if for all <span>( x, z in X)</span> and <i>y</i> on the path (in <i>T</i>) between <i>x</i> and <i>z</i>, we have <span>(textbf{d}(x, z) ge max {textbf{d}(x, y),)</span> <span>(textbf{d}(y, z))</span>}. If <i>T</i> is R-compatible with <span>((X, textbf{d}))</span>, then <i>T</i> is a minimum spanning of <span>((X, textbf{d}))</span>. We say that <span>((X, textbf{d}))</span> is <i>tree-Robinson</i> if all its minimum spanning trees are R-compatible. In this paper, we give a local characterization of these dissimilarities, in the sense that, although the definition of tree-Robinson dissimilarities involves all minimum spanning trees of <span>((X, textbf{d}))</span>, our characterization involves only some of them. This yields an efficient <span>(O(n^3))</span> algorithm to recognize these dissimilarities.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"30 3","pages":"849 - 862"},"PeriodicalIF":0.8,"publicationDate":"2026-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149007367","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}
Jesús A. De Loera, Carlos E. Valencia, Rafael H. Villarreal, Chengyang Wang
{"title":"Ehrhart Functions of Weighted Lattice Points","authors":"Jesús A. De Loera, Carlos E. Valencia, Rafael H. Villarreal, Chengyang Wang","doi":"10.1007/s00026-026-00822-0","DOIUrl":"10.1007/s00026-026-00822-0","url":null,"abstract":"<div><p>This paper studies three different ways to assign weights to the lattice points of a convex polytope and discusses the algebraic and combinatorial properties of the resulting weighted Ehrhart functions, their generating functions, and their associated rings. These will be called <i>q</i>-weighted, <i>r</i>-weighted, and <i>s</i>-weighted Ehrhart functions, respectively. The key questions we investigate are: <i>When are the weighted Ehrhart series rational functions, and which classical Ehrhart theory properties are preserved? And when are the abstract formal power series the Hilbert series of Ehrhart rings of some polytope?</i> We prove generalizations about weighted Ehrhart <span>(h^*)</span>-coefficients of <i>q</i>-weighted Ehrhart series and show <i>q</i>- and <i>s</i>-weighted Ehrhart reciprocity theorems. Then we show the <i>q</i>- and <i>r</i>-weighted Ehrhart rings are the (classical) Ehrhart rings of weight lifting polytopes.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"30 2","pages":"467 - 500"},"PeriodicalIF":0.8,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-026-00822-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148323472","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}
{"title":"Counting Permutations by Alternating Runs via Hetyei–Reiner Trees","authors":"Qiongqiong Pan, Yunze Wang, Jiang Zeng","doi":"10.1007/s00026-026-00817-x","DOIUrl":"10.1007/s00026-026-00817-x","url":null,"abstract":"<div><p>The generating polynomial of all <i>n</i>-permutations with respect to the number of alternating runs possesses a root at <span>(-1)</span> of multiplicity <span>(lfloor (n-2)/2 rfloor )</span> for <span>(n ge 2)</span>. This fact can be deduced by combining the David–Barton formula for Eulerian polynomials with the Foata–Schützenberger <span>(gamma )</span>-decomposition of these polynomials. Recently, Bóna provided a group—action proof of this result. In the present paper, we propose an alternative approach based on the Hetyei–Reiner action on binary trees, which yields a new combinatorial interpretation of Bóna’s quotient polynomial. Furthermore, we extend our study to analogous results for permutations of types B and D. As a consequence of our bijective framework, we also obtain combinatorial proofs of David–Barton type identities for permutations of types A and B.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"30 3","pages":"863 - 897"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149007374","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}
{"title":"Coalescence Probabilities of Cycle Products","authors":"Holden Mui","doi":"10.1007/s00026-026-00810-4","DOIUrl":"10.1007/s00026-026-00810-4","url":null,"abstract":"<div><p>Generalizing a formula of Stanley, we prove combinatorially that the probability that <span>(1, 2, dots , k)</span> are contained in the same cycle of a product of two random <i>n</i>-cycles is </p><div><div><span>$$frac{1}{k} + frac{4 (-1)^n}{ left( {begin{array}{c}2k kend{array}}right) } sum _{begin{array}{c} 1 le i le k-1 i not equiv n bmod 2 end{array}} left( {begin{array}{c}2k-1 k+iend{array}}right) left( frac{1}{n+i+1} - frac{1}{n-i}right) .$$</span></div></div></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"30 2","pages":"575 - 600"},"PeriodicalIF":0.8,"publicationDate":"2026-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148323218","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}
{"title":"A Polynomial Bosonic Form of Statistical Configuration Sums and the Odd/Even Minimal Excludant in Integer Partitions","authors":"Taichiro Takagi","doi":"10.1007/s00026-025-00792-9","DOIUrl":"10.1007/s00026-025-00792-9","url":null,"abstract":"<div><p>Inspired by the study of the minimal excludant in integer partitions by G.E. Andrews and D. Newman, we introduce a pair of new partition statistics, sqrank and rerank. They are related to a polynomial bosonic form of statistical configuration sums for an integrable cellular automaton. For all non-negative integers <i>n</i>, we prove that the partitions of <i>n</i> on which sqrank or rerank takes on a particular value, say <i>r</i>, are equinumerous with the partitions of <i>n</i> on which the odd/even minimal exclutant takes on the corresponding value, <span>(2r+1)</span> or <span>(2r+2)</span>.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"30 3","pages":"817 - 847"},"PeriodicalIF":0.8,"publicationDate":"2026-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-025-00792-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149007411","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}
Esther Banaian, Sunita Chepuri, Elizabeth Kelley, Sylvester W. Zhang
{"title":"Hydra Graphs for Graph LP Algebras","authors":"Esther Banaian, Sunita Chepuri, Elizabeth Kelley, Sylvester W. Zhang","doi":"10.1007/s00026-025-00797-4","DOIUrl":"10.1007/s00026-025-00797-4","url":null,"abstract":"<div><p>Graph LP algebras are a generalization of cluster algebras introduced by Lam and Pylyavskyy. We provide a combinatorial proof of positivity for certain cluster variables in these algebras. This proof uses a hypergraph generalization of snake graphs, a class of planar graphs which were used by Musiker, Schiffler, and Williams to prove positivity for cluster algebras from surfaces. These results extend those given in our previous paper, where we used a related combinatorial object known as a <i>T</i>-path.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"30 3","pages":"655 - 720"},"PeriodicalIF":0.8,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149007372","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}
{"title":"The Multiplicity of Eigenvalues of (A_{alpha }) of Trees","authors":"Zhibin Du","doi":"10.1007/s00026-025-00796-5","DOIUrl":"10.1007/s00026-025-00796-5","url":null,"abstract":"<div><p>Let <i>T</i> be a tree on <i>n</i> vertices. For <span>(alpha in [0,1),)</span> let <span>(A_{alpha }(T) = alpha D(T)+(1 - alpha ) A(T),)</span> where <i>D</i>(<i>T</i>) is the degree diagonal matrix of <i>T</i> and <i>A</i>(<i>T</i>) is the adjacency matrix of <i>T</i>. Denote by <span>(m_{A_{alpha }(T),theta })</span> the multiplicity of <span>(theta )</span> as an eigenvalue of <span>(A_{alpha }(T).)</span> When <span>(alpha = frac{1}{2})</span> (aiming to the Laplacian matrix of <i>T</i>), it is known in Akbari et al. (Linear Algebra Appl 586:262–273, 2020) that <span>(m_{A_{alpha }(T),theta } le frac{n - 3}{2},)</span> and when <span>(alpha = 0)</span> (aiming to the adjacency matrix of <i>T</i>), the upper bound of <span>(m_{A_{alpha }(T),theta })</span> for any fixed real number <span>(theta )</span> was recently obtained in Du and Huang (Linear Algebra Appl 654:56–68, 2022). In this paper, we establish an upper bound of <span>(m_{A_{alpha }(T),theta })</span> for any <span>(alpha in [0,1))</span> and any fixed real number <span>(theta ,)</span> unifying and generalizing the above-mentioned results about the multiplicities of eigenvalues of the adjacency matrix and Laplacian matrix of trees.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"30 3","pages":"945 - 964"},"PeriodicalIF":0.8,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149007440","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}
{"title":"Recursions for Multi-Component Extensions of the q-Dyson Constant Term Identity","authors":"Wenlong Jiang, Suzhen Wen, Yueming Zhong, Yue Zhou","doi":"10.1007/s00026-025-00790-x","DOIUrl":"10.1007/s00026-025-00790-x","url":null,"abstract":"<div><p>In this paper, we extend the <i>q</i>-Dyson constant term identity by categorizing the variables into <span>(p+1)</span> parts. The original <i>q</i>-Dyson constant term identity corresponds to the <span>(p=0)</span> case. Using Cai’s splitting approach, we establish a recursion for this constant term involving <span>(p+1)</span> components.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"30 3","pages":"721 - 736"},"PeriodicalIF":0.8,"publicationDate":"2025-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149007373","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}
Zeyu Huang, Timothy Huber, James McLaughlin, Pengjun Wang, Yan Xu, Dongxi Ye
{"title":"Sign Patterns of Certain Infinite Products","authors":"Zeyu Huang, Timothy Huber, James McLaughlin, Pengjun Wang, Yan Xu, Dongxi Ye","doi":"10.1007/s00026-025-00795-6","DOIUrl":"10.1007/s00026-025-00795-6","url":null,"abstract":"<div><p>The signs of Fourier coefficients of certain eta quotients are determined by dissecting expansions for theta functions and by applying a general dissection formula for certain classes of quintuple products. A characterization is given for the coefficient sign patterns for </p><div><div><span>$$begin{aligned} frac{(q^i;q^i)_{infty }}{(q^p;q^p)_{infty }} end{aligned}$$</span></div></div><p>for integers <span>( i > 1 )</span> and primes <span>( p > 3 )</span>. The sign analysis for this quotient addresses and extends a conjecture of Bringmann et al. for the coefficients of <span>( (q^2;q^2)_{infty }(q^5;q^5)_{infty }^{-1} )</span>. The sign distribution for additional classes of eta quotients is considered. This addresses multiple conjectures posed by Bringmann et al.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"30 3","pages":"919 - 943"},"PeriodicalIF":0.8,"publicationDate":"2025-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149007376","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}