European Journal of Combinatorics最新文献

{"title":"Strong parity edge-colorings of graphs","authors":"Peter Bradshaw , Sergey Norin , Douglas B. West","doi":"10.1016/j.ejc.2026.104343","DOIUrl":"10.1016/j.ejc.2026.104343","url":null,"abstract":"<div><div>An <em>edge-coloring</em> of a graph <span><math><mi>G</mi></math></span> assigns a color to each edge of <span><math><mi>G</mi></math></span>. An edge-coloring is a <em>parity edge-coloring</em> if for each path <span><math><mi>P</mi></math></span> in <span><math><mi>G</mi></math></span>, it uses some color on an odd number of edges in <span><math><mi>P</mi></math></span>. It is a <em>strong parity edge-coloring</em> if for every open walk <span><math><mi>W</mi></math></span> in <span><math><mi>G</mi></math></span>, it uses some color an odd number of times along <span><math><mi>W</mi></math></span>. The minimum numbers of colors in parity and strong parity edge-colorings of <span><math><mi>G</mi></math></span> are denoted <span><math><mrow><mi>p</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mover><mrow><mi>p</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, respectively.</div><div>We characterize strong parity edge-colorings and use this to prove lower bounds on <span><math><mrow><mover><mrow><mi>p</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and answer several questions of Bunde, Milans, West, and Wu. The applications are as follows. (1) We prove the conjecture that <span><math><mrow><mover><mrow><mi>p</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mi>s</mi><mo>∘</mo><mi>t</mi></mrow></math></span>, where <span><math><mrow><mi>s</mi><mo>∘</mo><mi>t</mi></mrow></math></span> is the Hopf–Stiefel function. (2) We show that <span><math><mrow><mover><mrow><mi>p</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for a connected <span><math><mi>n</mi></math></span>-vertex graph <span><math><mi>G</mi></math></span> equals the known lower bound <span><math><mrow><mo>⌈</mo><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi><mo>⌉</mo></mrow></math></span> if and only if <span><math><mi>G</mi></math></span> is a subgraph of the hypercube <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mrow><mo>⌈</mo><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi><mo>⌉</mo></mrow></mrow></msub></math></span>. (3) We asymptotically compute <span><math><mrow><mover><mrow><mi>p</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> when <span><math><mi>G</mi></math></span> is the <span><math><mi>ℓ</mi></math></span>th distance-power of a path, proving <span><math><mrow><mover><mrow><mi>p</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mrow><mo>(</mo><msubsup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>ℓ</mi></mrow></msubsup><mo>)</mo></mrow><mo>∼</mo><mi>ℓ</mi><mfenced><mrow><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></m","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104343"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078616","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}

{"title":"Necklaces, permutations, and periodic critical orbits for quadratic polynomials","authors":"Matthew Baker , Andrea Chen , Sophie Li , Matthew Qian","doi":"10.1016/j.ejc.2026.104352","DOIUrl":"10.1016/j.ejc.2026.104352","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> denote the <span><math><mi>n</mi></math></span>th <em>Gleason polynomial</em>, whose roots correspond to parameters <span><math><mi>c</mi></math></span> such that the critical point 0 is periodic of exact period <span><math><mi>n</mi></math></span> under iteration of <span><math><mrow><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>c</mi></mrow></math></span>, and let <span><math><msub><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub></math></span> denote the reduction of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> modulo 2. Buff, Floyd, Koch, and Parry made the surprising observation that the number of real roots of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is equal to the number of irreducible factors of <span><math><msub><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub></math></span> for all <span><math><mi>n</mi></math></span>. We provide a bijective proof for this result by first providing explicit bijections between (a) the set of real roots of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and the set <span><math><mrow><mover><mrow><mi>N</mi></mrow><mrow><mo>̄</mo></mrow></mover><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> of equivalence classes of primitive binary necklaces of length <span><math><mi>n</mi></math></span> under the inversion map swapping 0 and 1; and (b) the set of irreducible factors of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> modulo 2 and the set <span><math><mrow><msup><mrow><mover><mrow><mi>N</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow><mrow><mo>+</mo></mrow></msup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> of binary necklaces which are either primitive of length <span><math><mi>n</mi></math></span> with an even number of 1’s or primitive of length <span><math><mrow><mi>n</mi><mo>/</mo><mn>2</mn></mrow></math></span> with an odd number of 1’s. We then provide an explicit bijection, closely related to Milnor and Thurston’s kneading theory, between <span><math><mrow><mover><mrow><mi>N</mi></mrow><mrow><mo>̄</mo></mrow></mover><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msup><mrow><mover><mrow><mi>N</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow><mrow><mo>+</mo></mrow></msup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>. In addition, we provide explicit bijections between <span><math><mrow><mover><mrow><mi>N</mi></mrow><mrow><mo>̄</mo></mrow></mover><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>, the set <span><math><mrow><mi>CUP</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> ","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104352"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146189011","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}

{"title":"A weighted Murnaghan–Nakayama rule for (P,ω)-partitions","authors":"Per Alexandersson ,&nbsp;Olivia Nabawanda","doi":"10.1016/j.ejc.2025.104332","DOIUrl":"10.1016/j.ejc.2025.104332","url":null,"abstract":"<div><div>The <span><math><mrow><mo>(</mo><mi>P</mi><mo>,</mo><mi>ω</mi><mo>)</mo></mrow></math></span>-partition generating function <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mrow><mo>(</mo><mi>P</mi><mo>,</mo><mi>ω</mi><mo>)</mo></mrow></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> is a quasisymmetric function obtained from a labeled poset. Recently, Liu and Weselcouch gave a formula for the coefficients of <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mrow><mo>(</mo><mi>P</mi><mo>,</mo><mi>ω</mi><mo>)</mo></mrow></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> when expanded in the quasisymmetric power sum function basis. This formula generalizes the classical Murnaghan–Nakayama rule for Schur functions.</div><div>We extend this result to <em>weighted</em> <span><math><mrow><mo>(</mo><mi>P</mi><mo>,</mo><mi>ω</mi><mo>)</mo></mrow></math></span>-partitions and provide a short combinatorial proof, avoiding the Hopf algebra machinery used by Liu–Weselcouch.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104332"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978977","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}

{"title":"Majority bootstrap percolation on the permutahedron and other high-dimensional graphs","authors":"Maurício Collares ,&nbsp;Joshua Erde ,&nbsp;Anna Geisler ,&nbsp;Mihyun Kang","doi":"10.1016/j.ejc.2026.104347","DOIUrl":"10.1016/j.ejc.2026.104347","url":null,"abstract":"<div><div>Majority bootstrap percolation is a model of infection spreading in networks. Starting with a set of initially infected vertices, new vertices become infected once half of their neighbours are infected. Balogh, Bollobás and Morris studied this process on the hypercube and showed that there is a phase transition as the density of the initially infected set increases. Generalising their results to a broad class of high-dimensional graphs, the authors of this work established similar bounds on the critical window, establishing a universal behaviour for these graphs.</div><div>These methods necessitated an exponential bound on the order of the graphs in terms of their degrees. In this paper, we consider a slightly more restrictive class of high-dimensional graphs, which nevertheless covers most examples considered previously. Under these stronger assumptions, we are able to show that this universal behaviour holds in graphs of <em>superexponential order</em>. As a concrete and motivating example, we apply this result to the permutahedron, a symmetric high-dimensional graph of superexponential order which arises naturally in many areas of mathematics. Our methods also allow us to slightly improve the bounds on the critical window given in previous work, in particular in the case of the hypercube.</div><div>Finally, the upper and lower bounds on the critical window depend on the maximum and minimum degree of the graph, respectively, leading to much worse bounds for irregular graphs. We also analyse an explicit example of a high-dimensional <em>irregular</em> graph, namely the Cartesian product of stars and determine the first two terms in the expansion of the critical probability, which in this case is determined by the minimum degree.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104347"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146189012","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}

{"title":"On the anti-Ramsey threshold","authors":"Eden Kuperwasser","doi":"10.1016/j.ejc.2026.104344","DOIUrl":"10.1016/j.ejc.2026.104344","url":null,"abstract":"<div><div>We say that a graph <span><math><mi>G</mi></math></span> is anti-Ramsey for a graph <span><math><mi>H</mi></math></span> if any proper edge-colouring of <span><math><mi>G</mi></math></span> yields a rainbow copy of <span><math><mi>H</mi></math></span>, i.e. a copy of <span><math><mi>H</mi></math></span> whose edges all receive different colours. In this work we determine the threshold at which the binomial random graph becomes anti-Ramsey for any fixed graph <span><math><mi>H</mi></math></span>, given that <span><math><mi>H</mi></math></span> is sufficiently dense. Our proof employs a graph decomposition lemma in the style of the Nine Dragon Tree theorem, which may be of independent interest.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104344"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038268","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}

{"title":"Perfect matroids over skew hyperfields","authors":"Nathan Bowler ,&nbsp;Rudi Pendavingh","doi":"10.1016/j.ejc.2025.104327","DOIUrl":"10.1016/j.ejc.2025.104327","url":null,"abstract":"<div><div>A hyperfield <span><math><mi>H</mi></math></span> is <em>stringent</em> if <span><math><mrow><mi>a</mi><mo>⊞</mo><mi>b</mi></mrow></math></span> is a singleton unless <span><math><mrow><mi>a</mi><mo>=</mo><mo>−</mo><mi>b</mi></mrow></math></span>, for all <span><math><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>H</mi></mrow></math></span>. By a construction of Marc Krasner, each valued field gives rise to a stringent hyperfield. We show that if <span><math><mi>H</mi></math></span> is a stringent skew hyperfield, then weak matroids over <span><math><mi>H</mi></math></span> are strong matroids over <span><math><mi>H</mi></math></span>. Also, we present vector axioms for matroids over stringent skew hyperfields which generalize the vector axioms for oriented matroids and valuated matroids.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104327"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928549","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}

{"title":"Convergence of spectra of digraph limits","authors":"Jan Grebík ,&nbsp;Daniel Král’ ,&nbsp;Xizhi Liu ,&nbsp;Oleg Pikhurko ,&nbsp;Julia Slipantschuk","doi":"10.1016/j.ejc.2026.104349","DOIUrl":"10.1016/j.ejc.2026.104349","url":null,"abstract":"<div><div>The relation between densities of cycles and the spectrum of a graphon, which implies that the spectra of convergent graphons converge, fundamentally relies on the self-adjointness of the linear operator associated with a graphon. In this short paper, we consider the setting of digraphons, which are limits of directed graphs, and prove that the spectra of convergent digraphons converge. Using this result, we establish the relation between densities of directed cycles and the spectrum of a digraphon.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104349"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146189007","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}

{"title":"Recovery of cyclic words by their subwords","authors":"Sergey Luchinin ,&nbsp;Svetlana Puzynina ,&nbsp;Michaël Rao","doi":"10.1016/j.ejc.2025.104329","DOIUrl":"10.1016/j.ejc.2025.104329","url":null,"abstract":"<div><div>The problem of reconstructing words from their subwords involves determining the minimum amount of information needed, such as multisets of scattered subwords of a specific length or the frequency of scattered subwords from a given set, in order to uniquely identify a word. In this paper we show that a cyclic word on a binary alphabet can be reconstructed by its scattered subwords of length <span><math><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mi>n</mi><mo>+</mo><mn>4</mn></mrow></math></span>, and for each <span><math><mi>n</mi></math></span> one can find two cyclic words of length <span><math><mi>n</mi></math></span> which have the same set of scattered subwords of length <span><math><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mi>n</mi><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104329"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886335","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}

{"title":"Product representation of perfect cubes","authors":"Zsigmond György Fleiner ,&nbsp;Márk Hunor Juhász ,&nbsp;Blanka Kövér ,&nbsp;Péter Pál Pach ,&nbsp;Csaba Sándor","doi":"10.1016/j.ejc.2026.104342","DOIUrl":"10.1016/j.ejc.2026.104342","url":null,"abstract":"<div><div>Let <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> be the maximal size of a set <span><math><mrow><mi>A</mi><mo>⊆</mo><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow></math></span> such that the equation <span><span><span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo><mspace></mspace><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&lt;</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&lt;</mo><mo>⋯</mo><mo>&lt;</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span></span></span>has no solution with <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∈</mo><mi>A</mi></mrow></math></span> and integer <span><math><mi>x</mi></math></span>. Erdős, Sárközy and T. Sós studied <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow></msub></math></span>, and gave bounds when <span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn></mrow></math></span> and also in the general case. We study the problem for <span><math><mrow><mi>d</mi><mo>=</mo><mn>3</mn></mrow></math></span>, and provide bounds for <span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn></mrow></math></span> and 9, as well as in the general case. In particular, we refute an 18-year-old conjecture of Verstraëte.</div><div>We also introduce another function <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>d</mi></mrow></msub></math></span> closely related to <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>d</mi></mrow></msub></math></span>: While the original problem requires <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> to all be distinct, we can relax this and only require that the multiset of the <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>’s cannot be partitioned into <span><math><mi>d</mi></math></span>-tuples where each <span><math><mi>d</mi></math></span>-tuple consists of <span><math><mi>d</mi></math></span> copies of the same number.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104342"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038334","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}

{"title":"The partial derivative of ratios of Schur polynomials and applications to symplectic quotients","authors":"Hans-Christian Herbig ,&nbsp;Daniel Herden ,&nbsp;Harper Kolehmainen ,&nbsp;Christopher Seaton","doi":"10.1016/j.ejc.2026.104351","DOIUrl":"10.1016/j.ejc.2026.104351","url":null,"abstract":"<div><div>We show that a ratio of Schur polynomials <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>/</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>ρ</mi></mrow></msub></mrow></math></span> associated to partitions <span><math><mi>λ</mi></math></span> and <span><math><mi>ρ</mi></math></span> such that <span><math><mrow><mi>λ</mi><mo>⊊</mo><mi>ρ</mi></mrow></math></span> has a negative partial derivative at any point where all variables are positive. This is accomplished by establishing an injective map between sets of pairs of skew semistandard Young tableaux that preserves the product of the corresponding monomials. We use this result and the description of the first Laurent coefficient of the Hilbert series of the graded algebra of regular functions on a linear symplectic quotient by the circle to demonstrate that many such symplectic quotients are not graded regularly diffeomorphic. In addition, we give an upper bound for this Laurent coefficient in terms of the largest two weights of the circle representation and demonstrate that all but finitely many circle symplectic quotients of each dimension are not graded regularly diffeomorphic to linear symplectic quotients by <span><math><msub><mrow><mo>SU</mo></mrow><mrow><mn>2</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104351"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146189009","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}