{"title":"Combinatorial Properties of Three Classical Truncated Theta Series Theorems","authors":"Andrew Y. Z. Wang, Ang Xiao","doi":"10.1007/s00026-023-00658-y","DOIUrl":"10.1007/s00026-023-00658-y","url":null,"abstract":"<div><p>In this paper, we focus on the truncations of three classical theta series of Euler and Gauss, and analyze their combinatorial properties which play a key role in proving these truncated identities. Several interesting partition identities are established bijectively.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 2","pages":"681 - 699"},"PeriodicalIF":0.6,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45469629","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":"On the Compatible Sets Expansion of the Tutte Polynomial","authors":"Laura Pierson","doi":"10.1007/s00026-023-00657-z","DOIUrl":"10.1007/s00026-023-00657-z","url":null,"abstract":"<div><p>Kochol [6] gave a new expansion formula for the Tutte polynomial of a matroid using the notion of <i>compatible sets</i>, and asked how this expansion relates to the internal-external activities formula. Here, we provide an answer, which is obtained as a special case of a generalized version of the expansion formula to Las Vergnas’s trivariate Tutte polynomials of matroid perspectives [10]. The same generalization to matroid perspectives and bijection with activities have been independently proven by Kochol in [5] and [7] in parallel with this work, but using different methods. Kochol proves both results recursively using the contraction-deletion relations, whereas we give a more direct proof of the bijection and use that to deduce the compatible sets expansion formula from Las Vergnas’s activities expansion.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 1","pages":"33 - 42"},"PeriodicalIF":0.6,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47885699","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}
Yuliy Baryshnikov, Stephen Melczer, Robin Pemantle
{"title":"Asymptotics of Multivariate Sequences IV: Generating Functions with Poles on a Hyperplane Arrangement","authors":"Yuliy Baryshnikov, Stephen Melczer, Robin Pemantle","doi":"10.1007/s00026-023-00654-2","DOIUrl":"10.1007/s00026-023-00654-2","url":null,"abstract":"<div><p>Let <span>(F(z_1,dots ,z_d))</span> be the quotient of an analytic function with a product of linear functions. Working in the framework of analytic combinatorics in several variables, we compute asymptotic formulae for the Taylor coefficients of <i>F</i> using multivariate residues and saddle-point approximations. Because the singular set of <i>F</i> is the union of hyperplanes, we are able to make explicit the topological decompositions which arise in the multivariate singularity analysis. In addition to effective and explicit asymptotic results, we provide the first results on transitions between different asymptotic regimes, and provide the first software package to verify and compute asymptotics in non-smooth cases of analytic combinatorics in several variables. It is also our hope that this paper will serve as an entry to the more advanced corners of analytic combinatorics in several variables for combinatorialists.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 1","pages":"169 - 221"},"PeriodicalIF":0.6,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136066722","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}
Katharina T. Huber, Vincent Moulton, Megan Owen, Andreas Spillner, Katherine St. John
{"title":"The Space of Equidistant Phylogenetic Cactuses","authors":"Katharina T. Huber, Vincent Moulton, Megan Owen, Andreas Spillner, Katherine St. John","doi":"10.1007/s00026-023-00656-0","DOIUrl":"10.1007/s00026-023-00656-0","url":null,"abstract":"<div><p>An <i>equidistant</i> <i>X</i>-<i>cactus</i> is a type of rooted, arc-weighted, directed acyclic graph with leaf set <i>X</i>, that is used in biology to represent the evolutionary history of a set <span>(X)</span> of species. In this paper, we introduce and investigate the space of equidistant <i>X</i>-cactuses. This space contains, as a subset, the space of ultrametric trees on <i>X</i> that was introduced by Gavryushkin and Drummond. We show that equidistant-cactus space is a CAT(0)-metric space which implies, for example, that there are unique geodesic paths between points. As a key step to proving this, we present a combinatorial result concerning <i>ranked</i> rooted <i>X</i>-cactuses. In particular, we show that such graphs can be encoded in terms of a pairwise compatibility condition arising from a poset of collections of pairs of subsets of <span>(X)</span> that satisfy certain set-theoretic properties. As a corollary, we also obtain an encoding of ranked, rooted <i>X</i>-trees in terms of partitions of <i>X</i>, which provides an alternative proof that the space of ultrametric trees on <i>X</i> is CAT(0). We expect that our results will provide the basis for novel ways to perform statistical analyses on collections of equidistant <i>X</i>-cactuses, as well as new directions for defining and understanding spaces of more general, arc-weighted phylogenetic networks.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 1","pages":"1 - 32"},"PeriodicalIF":0.6,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10904525/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45787472","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":"The Passing of Marko Petkovšek","authors":"Andrej Bauer, Sandi Klavžar","doi":"10.1007/s00026-023-00653-3","DOIUrl":"10.1007/s00026-023-00653-3","url":null,"abstract":"","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"27 2","pages":"455 - 456"},"PeriodicalIF":0.5,"publicationDate":"2023-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48095647","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":"On Discrete Gradient Vector Fields and Laplacians of Simplicial Complexes","authors":"Ivan Contreras, Andrew Tawfeek","doi":"10.1007/s00026-023-00655-1","DOIUrl":"10.1007/s00026-023-00655-1","url":null,"abstract":"<div><p>Discrete Morse theory, a cell complex-analog to smooth Morse theory allowing homotopic tools in the discrete realm, has been developed over the past few decades since its original formulation by Robin Forman in 1998. In particular, discrete gradient vector fields on simplicial complexes capture important topological features of the structure. We prove that the characteristic polynomials of the Laplacian matrices of a simplicial complex are generating functions for discrete gradient vector fields if the complex is a triangulation of an orientable manifold. Furthermore, we provide a full characterization of the correspondence between rooted forests in higher dimensions and discrete gradient vector fields.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 1","pages":"67 - 91"},"PeriodicalIF":0.6,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135478839","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":"Partial Symmetries of Iterated Plethysms","authors":"Álvaro Gutiérrez, Mercedes H. Rosas","doi":"10.1007/s00026-023-00652-4","DOIUrl":"10.1007/s00026-023-00652-4","url":null,"abstract":"<div><p>This work highlights the existence of partial symmetries in large families of iterated plethystic coefficients. The plethystic coefficients involved come from the expansion in the Schur basis of iterated plethysms of Schur functions indexed by one-row partitions.The partial symmetries are described in terms of an involution on partitions, the flip involution, that generalizes the ubiquitous <span>(omega )</span> involution. Schur-positive symmetric functions possessing this partial symmetry are termed flip-symmetric. The operation of taking plethysm with <span>(s_lambda )</span> preserves flip-symmetry, provided that <span>(lambda )</span> is a partition of two. Explicit formulas for the iterated plethysms <span>(s_2circ s_bcirc s_a)</span> and <span>(s_ccirc s_2circ s_a)</span>, with <i>a</i>, <i>b</i>, and <i>c</i> <span>(ge )</span> 2 allow us to show that these two families of iterated plethysms are flip-symmetric. The article concludes with some observations, remarks, and open questions on the unimodality and asymptotic normality of certain flip-symmetric sequences of iterated plethystic coefficients.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"27 3","pages":"493 - 518"},"PeriodicalIF":0.5,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49572412","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":"Folding Rotationally Symmetric Tableaux via Webs","authors":"Kevin Purbhoo, Shelley Wu","doi":"10.1007/s00026-023-00648-0","DOIUrl":"10.1007/s00026-023-00648-0","url":null,"abstract":"<div><p>Rectangular standard Young tableaux with 2 or 3 rows are in bijection with <span>(U_q(mathfrak {sl}_2))</span>-webs and <span>(U_q(mathfrak {sl}_3))</span>-webs, respectively. When <span>(mathcal {W})</span> is a web with a reflection symmetry, the corresponding tableau <span>(T_mathcal {W})</span> has a rotational symmetry. Folding <span>(T_mathcal {W})</span> transforms it into a domino tableau <span>(D_mathcal {W})</span>. We study the relationships between these correspondences. For 2-row tableaux, folding a rotationally symmetric tableau corresponds to “literally folding” the web along its axis of symmetry. For 3-row tableaux, we give simple algorithms, which provide direct bijective maps between symmetrical webs and domino tableaux (in both directions). These details of these algorithms reflect the intuitive idea that <span>(D_mathcal {W})</span> corresponds to “<span>(mathcal {W})</span> modulo symmetry”.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 1","pages":"93 - 119"},"PeriodicalIF":0.6,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49240960","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}
Andrzej Dudek, Jarosław Grytczuk, Andrzej Ruciński
{"title":"Long Twins in Random Words","authors":"Andrzej Dudek, Jarosław Grytczuk, Andrzej Ruciński","doi":"10.1007/s00026-023-00651-5","DOIUrl":"10.1007/s00026-023-00651-5","url":null,"abstract":"<div><p><i>Twins</i> in a finite word are formed by a pair of identical subwords placed at disjoint sets of positions. We investigate the maximum length of twins in <i>a random</i> word over a <i>k</i>-letter alphabet. The obtained lower bounds for small values of <i>k</i> significantly improve the best estimates known in the deterministic case. Bukh and Zhou in 2016 showed that every ternary word of length <i>n</i> contains twins of length at least 0.34<i>n</i>. Our main result states that in a random ternary word of length <i>n</i>, with high probability, one can find twins of length at least 0.41<i>n</i>. In the general case of alphabets of size <span>(kgeqslant 3)</span> we obtain analogous lower bounds of the form <span>(frac{1.64}{k+1}n)</span> which are better than the known deterministic bounds for <span>(kleqslant 354)</span>. In addition, we present similar results for <i>multiple</i> twins in random words.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"27 3","pages":"749 - 768"},"PeriodicalIF":0.5,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-023-00651-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46164776","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":"On the Subdivision Algebra for the Polytope (mathcal {U}_{I,overline{J}})","authors":"Matias von Bell, Martha Yip","doi":"10.1007/s00026-023-00650-6","DOIUrl":"10.1007/s00026-023-00650-6","url":null,"abstract":"<div><p>The polytopes <span>(mathcal {U}_{I,overline{J}})</span> were introduced by Ceballos, Padrol, and Sarmiento to provide a geometric approach to the study of <span>((I,overline{J}))</span>-Tamari lattices. They observed a connection between certain <span>(mathcal {U}_{I,overline{J}})</span> and acyclic root polytopes, and wondered if Mészáros’ subdivision algebra can be used to subdivide all <span>(mathcal {U}_{I,overline{J}})</span>. We answer this in the affirmative from two perspectives, one using flow polytopes and the other using root polytopes. We show that <span>(mathcal {U}_{I,overline{J}})</span> is integrally equivalent to a flow polytope that can be subdivided using the subdivision algebra. Alternatively, we find a suitable projection of <span>(mathcal {U}_{I,overline{J}})</span> to an acyclic root polytope which allows subdivisions of the root polytope to be lifted back to <span>(mathcal {U}_{I,overline{J}})</span>. As a consequence, this implies that subdivisions of <span>(mathcal {U}_{I,overline{J}})</span> can be obtained with the algebraic interpretation of using reduced forms of monomials in the subdivision algebra. In addition, we show that the <span>((I,overline{J}))</span>-Tamari complex can be obtained as a triangulated flow polytope.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 1","pages":"43 - 65"},"PeriodicalIF":0.6,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45728539","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}