Pub Date : 2019-01-26DOI: 10.4310/JOC.2019.v10.n1.a1
R. Yuster
Erdős and Hanani proved that for every fixed integer $k ge 2$, the complete graph $K_n$ can be almost completely packed with copies of $K_k$; that is, $K_n$ contains pairwise edge-disjoint copies of $K_k$ that cover all but an $o_n(1)$ fraction of its edges. Equivalently, elements of the set $C(k)$ of all red-blue edge colorings of $K_k$ can be used to almost completely pack every red-blue edge coloring of $K_n$. The following strengthening of the aforementioned Erdős-Hanani result is considered. Suppose $C' subset C(k)$. Is it true that we can use elements only from $C'$ and almost completely pack every red-blue edge coloring of $K_n$? An element $C in C(k)$ is {em avoidable} if $C'=C(k) setminus C$ has this property and a subset ${cal F} subset C(k)$ is avoidable if $C'=C(k) setminus {cal F}$ has this property. It seems difficult to determine all avoidable graphs as well as all avoidable families. We prove some nontrivial sufficient conditions for avoidability. Our proofs imply, in particular, that (i) almost all elements of $C(k)$ are avoidable (ii) all Eulerian elements of $C(k)$ are avoidable and, in fact, the set of all Eulerian elements of $C(k)$ is avoidable.
{"title":"Packing without some pieces","authors":"R. Yuster","doi":"10.4310/JOC.2019.v10.n1.a1","DOIUrl":"https://doi.org/10.4310/JOC.2019.v10.n1.a1","url":null,"abstract":"Erdős and Hanani proved that for every fixed integer $k ge 2$, the complete graph $K_n$ can be almost completely packed with copies of $K_k$; that is, $K_n$ contains pairwise edge-disjoint copies of $K_k$ that cover all but an $o_n(1)$ fraction of its edges. Equivalently, elements of the set $C(k)$ of all red-blue edge colorings of $K_k$ can be used to almost completely pack every red-blue edge coloring of $K_n$. \u0000The following strengthening of the aforementioned Erdős-Hanani result is considered. Suppose $C' subset C(k)$. Is it true that we can use elements only from $C'$ and almost completely pack every red-blue edge coloring of $K_n$? An element $C in C(k)$ is {em avoidable} if $C'=C(k) setminus C$ has this property and a subset ${cal F} subset C(k)$ is avoidable if $C'=C(k) setminus {cal F}$ has this property. \u0000It seems difficult to determine all avoidable graphs as well as all avoidable families. We prove some nontrivial sufficient conditions for avoidability. Our proofs imply, in particular, that (i) almost all elements of $C(k)$ are avoidable (ii) all Eulerian elements of $C(k)$ are avoidable and, in fact, the set of all Eulerian elements of $C(k)$ is avoidable.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"84 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85515239","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}
Pub Date : 2019-01-06DOI: 10.4310/joc.2020.v11.n3.a1
Brittney Ellzey, M. Wachs
A Smirnov word is a word over the positive integers in which adjacent letters must be different. A symmetric function enumerating these words by descent number arose in the work of Shareshian and the second named author on $q$-Eulerian polynomials, where a $t$-analog of a formula of Carlitz, Scoville, and Vaughan for enumerating Smirnov words is proved. A symmetric function enumerating a circular version of these words by cyclic descent number arose in the work of the first named author on chromatic quasisymmetric functions of directed graphs, where a $t$-analog of a formula of Stanley for enumerating circular Smirnov words is proved. In this paper we obtain new $t$-analogs of the Carlitz-Scoville-Vaughan formula and the Stanley formula in which the roles of descent number and cyclic descent number are switched. These formulas show that the Smirnov word enumerators are polynomials in $t$ whose coefficients are e-positive symmetric functions. We also obtain expansions in the power sum basis and the fundamental quasisymmetric function basis, complementing earlier results of Shareshian and the authors. Our work relies on studying refinements of the Smirnov word enumerators that count certain restricted classes of Smirnov words by descent number. Applications to variations of $q$-Eulerian polynomials and to the chromatic quasisymmetric functions introduced by Shareshian and the second named author are also presented.
{"title":"On enumerators of Smirnov words by descents and cyclic descents","authors":"Brittney Ellzey, M. Wachs","doi":"10.4310/joc.2020.v11.n3.a1","DOIUrl":"https://doi.org/10.4310/joc.2020.v11.n3.a1","url":null,"abstract":"A Smirnov word is a word over the positive integers in which adjacent letters must be different. A symmetric function enumerating these words by descent number arose in the work of Shareshian and the second named author on $q$-Eulerian polynomials, where a $t$-analog of a formula of Carlitz, Scoville, and Vaughan for enumerating Smirnov words is proved. A symmetric function enumerating a circular version of these words by cyclic descent number arose in the work of the first named author on chromatic quasisymmetric functions of directed graphs, where a $t$-analog of a formula of Stanley for enumerating circular Smirnov words is proved. \u0000In this paper we obtain new $t$-analogs of the Carlitz-Scoville-Vaughan formula and the Stanley formula in which the roles of descent number and cyclic descent number are switched. These formulas show that the Smirnov word enumerators are polynomials in $t$ whose coefficients are e-positive symmetric functions. We also obtain expansions in the power sum basis and the fundamental quasisymmetric function basis, complementing earlier results of Shareshian and the authors. \u0000Our work relies on studying refinements of the Smirnov word enumerators that count certain restricted classes of Smirnov words by descent number. Applications to variations of $q$-Eulerian polynomials and to the chromatic quasisymmetric functions introduced by Shareshian and the second named author are also presented.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"17 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83550274","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}
Pub Date : 2019-01-01DOI: 10.1007/978-3-030-00831-4_8
P. Mladenovic
{"title":"Graph Theory: Part 1","authors":"P. Mladenovic","doi":"10.1007/978-3-030-00831-4_8","DOIUrl":"https://doi.org/10.1007/978-3-030-00831-4_8","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"20 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82736695","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}
Pub Date : 2019-01-01DOI: 10.1007/978-3-030-00831-4_2
P. Mladenovic
{"title":"Arrangements, Permutations, and Combinations","authors":"P. Mladenovic","doi":"10.1007/978-3-030-00831-4_2","DOIUrl":"https://doi.org/10.1007/978-3-030-00831-4_2","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"16 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86611611","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}
Pub Date : 2019-01-01DOI: 10.1007/978-3-030-00831-4_10
P. Mladenovic
{"title":"Existence of Combinatorial Configurations","authors":"P. Mladenovic","doi":"10.1007/978-3-030-00831-4_10","DOIUrl":"https://doi.org/10.1007/978-3-030-00831-4_10","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"21 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86020349","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}
Pub Date : 2019-01-01DOI: 10.4310/JOC.2019.V10.N4.A3
A. Garsia, J. Haglund, Marino Romero
We prove a technical identity involving the Δ operator from Macdonald polynomial theory, which allows us to transform expressions involving the Δ operator and the Hall scalar product into other such expressions. We show how our technical identity, although following easily from the well-known Koornwinder-Macdonald reciprocity theorem, contains as special cases several identities occur-ing in the literature, proved there by more complicated arguments. We also show how our identity can be used to obtain some new expressions for the q, t -Narayana numbers introduced by Dukes and Le Borgne, as well as new identities involving the Δ operator and the power sum symmetric function p n .
我们证明了麦克唐纳多项式理论中涉及Δ算子的技术恒等式，它允许我们将涉及Δ算子和Hall标量积的表达式转换为其他此类表达式。我们展示了我们的技术恒等式，虽然很容易从著名的Koornwinder-Macdonald互易定理中得到，但作为特例，它包含了文献中出现的几个恒等式，这些恒等式是通过更复杂的论证证明的。我们还展示了如何使用我们的恒等式来获得Dukes和Le Borgne引入的q, t -Narayana数的一些新表达式，以及涉及Δ算子和幂和对称函数pn的新恒等式。
{"title":"Some new symmetric function tools and their applications","authors":"A. Garsia, J. Haglund, Marino Romero","doi":"10.4310/JOC.2019.V10.N4.A3","DOIUrl":"https://doi.org/10.4310/JOC.2019.V10.N4.A3","url":null,"abstract":"We prove a technical identity involving the Δ operator from Macdonald polynomial theory, which allows us to transform expressions involving the Δ operator and the Hall scalar product into other such expressions. We show how our technical identity, although following easily from the well-known Koornwinder-Macdonald reciprocity theorem, contains as special cases several identities occur-ing in the literature, proved there by more complicated arguments. We also show how our identity can be used to obtain some new expressions for the q, t -Narayana numbers introduced by Dukes and Le Borgne, as well as new identities involving the Δ operator and the power sum symmetric function p n .","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"48 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91169353","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}
Pub Date : 2019-01-01DOI: 10.4310/JOC.2019.V10.N3.A6
S. Williamson
In their paper, Large-scale regularities of lattice embeddings of posets, Remmel and Williamson study posets and their incompa-rability graphs on N k . Properties (1) through (3) of their main result, Theorem 1.5, are proved using Ramsey theory. The proof of Theorem 1.5 (4), however, uses Friedman’s Jump Free Theorem, a powerful ZFC independent extension of Ramsey theory. Attempts to prove Theorem 1.5 (4) within the ZFC axioms have thus far failed. This leaves the main result of the Remmel-Williamson paper in what we informally call “ZFC limbo.” In this paper we explore other results of this type. In particular, Theorem 6.2 of this paper, which we prove to be independent of ZFC, directly implies our very similar Theorem 6.3 for which we have no ZFC proof. On the basis of the close structural similarity between these two theorems, we conjecture that Theorem 6.3 is also independent of ZFC. However, Theorem 6.3 also follows directly from “subset sum is solvable in polynomial time.” Of course, if our conjecture is true, “subset sum is solvable in polynomial time” cannot be proved in ZFC.
{"title":"Combinatorics in ZFC limbo","authors":"S. Williamson","doi":"10.4310/JOC.2019.V10.N3.A6","DOIUrl":"https://doi.org/10.4310/JOC.2019.V10.N3.A6","url":null,"abstract":"In their paper, Large-scale regularities of lattice embeddings of posets, Remmel and Williamson study posets and their incompa-rability graphs on N k . Properties (1) through (3) of their main result, Theorem 1.5, are proved using Ramsey theory. The proof of Theorem 1.5 (4), however, uses Friedman’s Jump Free Theorem, a powerful ZFC independent extension of Ramsey theory. Attempts to prove Theorem 1.5 (4) within the ZFC axioms have thus far failed. This leaves the main result of the Remmel-Williamson paper in what we informally call “ZFC limbo.” In this paper we explore other results of this type. In particular, Theorem 6.2 of this paper, which we prove to be independent of ZFC, directly implies our very similar Theorem 6.3 for which we have no ZFC proof. On the basis of the close structural similarity between these two theorems, we conjecture that Theorem 6.3 is also independent of ZFC. However, Theorem 6.3 also follows directly from “subset sum is solvable in polynomial time.” Of course, if our conjecture is true, “subset sum is solvable in polynomial time” cannot be proved in ZFC.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"51 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90807928","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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