{"title":"Interview with Benny Sudakov","authors":"","doi":"10.54550/eca2024v4s3i3","DOIUrl":"https://doi.org/10.54550/eca2024v4s3i3","url":null,"abstract":"","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135617423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Richard Ehrenborg, Gábor Hetyei, Margaret A. Readdy
{"title":"Catalan--Spitzer permutations","authors":"Richard Ehrenborg, Gábor Hetyei, Margaret A. Readdy","doi":"10.54550/eca2024v4s2r15","DOIUrl":"https://doi.org/10.54550/eca2024v4s2r15","url":null,"abstract":"We study two classes of permutations intimately related to the visual proof of Spitzer's lemma and Huq's generalization of the Chung-Feller theorem. Both classes of permutations are counted by the Fuss-Catalan numbers. The study of one class leads to a generalization of results of Flajolet from continued fractions to continuants. The study of the other class leads to the discovery of a restricted variant of the Foata--Strehl group action.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139334080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"New proofs of interlacing of zeros of Eulerian polynomials. III","authors":"Chak-On Chow","doi":"10.54550/eca2024v4s2r14","DOIUrl":"https://doi.org/10.54550/eca2024v4s2r14","url":null,"abstract":": Many generating functions of combinatorial systems have palindromic coefficients. A notable example is the n th Eulerian polynomial A n ( x ). It is known that a palindromic polynomial f ( x ) of degree 2 n can be expressed as x n Q ( x + 1 x ) for some polynomial Q ( x ) of degree n . By exploring the real-rootedness of Q ( x ), we are able to infer the corresponding property of f ( x ). By representing A n ( x ) in the said form, we give new proof of the real-rootedness and interlacing property of A n ( x ). This same approach applied to the n th alternating Eulerian polynomial (cid:98) A n ( x ) allows us to infer the interlacing/alternating property of the real and imaginary parts of its non-real zeros. The analogous type B results are also presented.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"238 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139334552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Interview with David Conlon","authors":"Toufik Mansour, Elaine Doyle","doi":"10.54550/eca2024v4s2i5","DOIUrl":"https://doi.org/10.54550/eca2024v4s2i5","url":null,"abstract":"","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139334083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Shiliang Gao, Joshua Kiers, Gidon Orelowitz, Alexander Yong
{"title":"The Kostka semigroup and its Hilbert basis","authors":"Shiliang Gao, Joshua Kiers, Gidon Orelowitz, Alexander Yong","doi":"10.54550/eca2024v4s2r9","DOIUrl":"https://doi.org/10.54550/eca2024v4s2r9","url":null,"abstract":"The Kostka semigroup consists of pairs of partitions with at most r parts that have positive Kostka coefficient. For this semigroup, Hilbert basis membership is an NP-complete problem. We introduce KGR graphs and conservative subtrees, through the Gale-Ryser theorem on contingency tables, as a criterion for membership. In our main application, we show that if a partition pair is in the Hilbert basis then the partitions are at most r wide. We also classify the extremal rays of the associated polyhedral cone; these rays correspond to a (strict) subset of the Hilbert basis. In an appendix, the second and third authors show that a natural extension of our main result on the Kostka semigroup cannot be extended to the Littlewood-Richardson semigroup. This furthermore gives a counterexample to a recent speculation of P. Belkale concerning the semigroup controlling nonvanishing conformal blocks.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135131269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Interview with Ken-ichi Kawarabayashi","authors":"","doi":"10.54550/eca2024v4s2i2","DOIUrl":"https://doi.org/10.54550/eca2024v4s2i2","url":null,"abstract":"","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135295739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the formal power series of involutory functions","authors":"Alfred Schreiber","doi":"10.54550/eca2024v4s2r10","DOIUrl":"https://doi.org/10.54550/eca2024v4s2r10","url":null,"abstract":": It is shown that the coefficients of any involutory function f represented as a power series can be expressed in terms of multivariable Lah polynomials. This result is based on the fact that any such f ( (cid:54) = identity) can be regarded as a (compositional) conjugate of negative identity. Moreover, a constructive proof of this statement is given","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135295738","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Interview with Alan Frieze","authors":"Toufik Mansour","doi":"10.54550/eca2024v4s2i4","DOIUrl":"https://doi.org/10.54550/eca2024v4s2i4","url":null,"abstract":"","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139334435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hemanshu Kaul, Michael Maxfield, Jeffrey Mudrock, Seth Thomason
{"title":"The DP color function of clique-gluings of graphs","authors":"Hemanshu Kaul, Michael Maxfield, Jeffrey Mudrock, Seth Thomason","doi":"10.54550/eca2024v4s2r11","DOIUrl":"https://doi.org/10.54550/eca2024v4s2r11","url":null,"abstract":"DP-coloring (also called correspondence coloring) is a generalization of list coloring that has been widely studied in recent years after its introduction by Dvov{r}'{a}k and Postle in 2015. As the analogue of the chromatic polynomial of a graph $G$, $P(G,m)$, the DP color function of $G$, denoted by $P_{DP}(G,m)$, counts the minimum number of DP-colorings over all possible $m$-fold covers. Formulas for chromatic polynomials of clique-gluings of graphs, a fundamental graph operation, are well-known, but the effect of such gluings on the DP color function is not well understood. In this paper we study the DP color function of $K_p$-gluings of graphs. Recently, Becker et. al. asked whether $P_{DP}(G,m) leq (prod_{i=1}^n P_{DP}(G_i,m))/left( prod_{i=0}^{p-1} (m-i) right)^{n-1}$ whenever $m geq p$, where the expression on the right is the DP-coloring analogue of the corresponding chromatic polynomial formula for a $K_p$-gluing, $G$, of $G_1, ldots, G_n$. Becker et. al. showed this inequality holds when $p=1$. In this paper we show this inequality holds for edge-gluings ($p=2$). On the other hand, we show it does not hold for triangle-gluings ($p=3$), which also answers a question of Dong and Yang (2021). Finally, we show a relaxed version, based on a class of $m$-fold covers that we conjecture would yield the fewest DP-colorings for a given graph, of the inequality holds when $p geq 3$.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"37 5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135296696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the homology of several number-theoretic set families","authors":"Marcel Goh, Jonah Saks","doi":"10.54550/eca2024v4s2r12","DOIUrl":"https://doi.org/10.54550/eca2024v4s2r12","url":null,"abstract":"This paper describes the homology of various simplicial complexes associated to set families from combinatorial number theory, including primitive sets, pairwise coprime sets, product-free sets, and coprime-free sets. We present a condition on a set family that results in easy computation of the homology groups, and show that the first three examples, among many others, admit such a structure. We then extend our techniques to address the complexes associated to coprime-free sets and a generalization of primitive sets.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135343380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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