We say that two permutations $pi$ and $rho$ have separated descents at position $k$ if $pi$ has no descents before position $k$ and $rho$ has no descents after position $k$. We give a counting formula, in terms of reduced word tableaux, for computing the structure constants of products of Schubert polynomials indexed by permutations with separated descents. Our approach uses generalizations of Sch"utzenberger's jeu de taquin algorithm and the Edelman-Greene correspondence via bumpless pipe dreams.
如果$pi$在位置$k$之前没有下降，并且$rho$在位置$k$之后没有下降，那么我们说两个排列$pi$和$rho$在位置$k$处有分开的下降。我们给出了一个简化词表的计算公式，用于计算舒伯特多项式乘积的结构常数，这些乘积是由具有分离下降的排列索引的。我们的方法使用了sch曾伯格的jeu de taquin算法和Edelman-Greene对应通过无碰撞白日梦的推广。
{"title":"Schubert Products for Permutations with Separated Descents.","authors":"Daoji Huang","doi":"10.1093/imrn/rnac299","DOIUrl":"https://doi.org/10.1093/imrn/rnac299","url":null,"abstract":"We say that two permutations $pi$ and $rho$ have separated descents at position $k$ if $pi$ has no descents before position $k$ and $rho$ has no descents after position $k$. We give a counting formula, in terms of reduced word tableaux, for computing the structure constants of products of Schubert polynomials indexed by permutations with separated descents. Our approach uses generalizations of Sch\"utzenberger's jeu de taquin algorithm and the Edelman-Greene correspondence via bumpless pipe dreams.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78853390","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 : 2020-12-14DOI: 10.29020/NYBG.EJPAM.V14I1.3900
R. Corcino, Jay M. Ontolan, Maria Rowena S. Lobrigas
In this paper, a q-analogue of r-Whitney-Lah numbers, also known as (q,r)-Whitney-Lah number, denoted by $L_{m,r}[n,k]_q$ is defined using the triangular recurrence relation. Several fundamental properties for the q-analogue are established such as vertical and horizontal recurrence relations, horizontal and exponential generating functions. Moreover, an explicit formula for (q,r)-Whitney-Lah number is derived using the concept of q-difference operator, particularly, the q-analogue of Newton's Interpolation Formula (the umbral version of Taylor series). Furthermore, an explicit formula for the first form (q,r)-Dowling numbers is obtained which is expressed in terms of (q,r)-Whitney-Lah numbers and (q,r)-Whitney numbers of the second kind.
{"title":"Explicit Formulas for the First Form (q,r)-Dowling Numbers and (q,r)-Whitney-Lah Numbers","authors":"R. Corcino, Jay M. Ontolan, Maria Rowena S. Lobrigas","doi":"10.29020/NYBG.EJPAM.V14I1.3900","DOIUrl":"https://doi.org/10.29020/NYBG.EJPAM.V14I1.3900","url":null,"abstract":"In this paper, a q-analogue of r-Whitney-Lah numbers, also known as (q,r)-Whitney-Lah number, denoted by $L_{m,r}[n,k]_q$ is defined using the triangular recurrence relation. Several fundamental properties for the q-analogue are established such as vertical and horizontal recurrence relations, horizontal and exponential generating functions. Moreover, an explicit formula for (q,r)-Whitney-Lah number is derived using the concept of q-difference operator, particularly, the q-analogue of Newton's Interpolation Formula (the umbral version of Taylor series). Furthermore, an explicit formula for the first form (q,r)-Dowling numbers is obtained which is expressed in terms of (q,r)-Whitney-Lah numbers and (q,r)-Whitney numbers of the second kind.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90312611","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}
We introduce the concept of deformed zero-determinant strategies in repeated games. We then show that the Tit-for-Tat strategy in the repeated prisoner's dilemma game is a deformed zero-determinant strategy, which unilaterally equalizes the probability distribution functions of payoffs of two players.
{"title":"Tit-for-Tat Strategy as a Deformed Zero-Determinant Strategy in Repeated Games","authors":"M. Ueda","doi":"10.7566/JPSJ.90.025002","DOIUrl":"https://doi.org/10.7566/JPSJ.90.025002","url":null,"abstract":"We introduce the concept of deformed zero-determinant strategies in repeated games. We then show that the Tit-for-Tat strategy in the repeated prisoner's dilemma game is a deformed zero-determinant strategy, which unilaterally equalizes the probability distribution functions of payoffs of two players.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86992966","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 : 2020-12-07DOI: 10.1016/J.JNT.2021.02.011
J. Guo
{"title":"An inequality for coefficients of the real-rooted polynomials","authors":"J. Guo","doi":"10.1016/J.JNT.2021.02.011","DOIUrl":"https://doi.org/10.1016/J.JNT.2021.02.011","url":null,"abstract":"","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78099644","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 : 2020-11-30DOI: 10.2140/involve.2022.15.89
V. Bergelson, Jake Huryn, R. Raghavan
We explore the properties of non-piecewise syndetic sets with positive upper density, which we call discordant, in countable amenable (semi)groups. Sets of this kind are involved in many questions of Ramsey theory and manifest the difference in complexity between the classical van der Waerden's theorem and Szemer'edi's theorem. We generalize and unify old constructions and obtain new results about these historically interesting sets. Here is a small sample of our results. $bullet$ We connect discordant sets to recurrence in dynamical systems, and in this setting we exhibit an intimate analogy between discordant sets and nowhere dense sets having positive measure. $bullet$ We introduce a wide-ranging generalization of the squarefree numbers, producing many examples of discordant sets in $mathbb{Z}$, $mathbb{Z}^d$, and the Heisenberg group. We develop a unified method to compute densities of these discordant sets. $bullet$ We show that, for any countable abelian group $G$, any F{o}lner sequence $Phi$ in $G$, and any $c in (0, 1)$, there exists a discordant set $A subseteq G$ with $d_Phi(A) = c$. Here $d_Phi$ denotes density along $Phi$. Along the way, we draw from various corners of mathematics, including classical Ramsey theory, ergodic theory, number theory, and topological and symbolic dynamics.
研究了可数可调半群中具有正上密度的非分段合成集的性质，我们称之为不协调集。这类集合涉及到拉姆齐理论的许多问题，并体现了经典的范德威登定理和塞默迪定理在复杂性上的不同。我们对这些历史上有趣的集合进行了推广和统一，并得到了新的结果。这是我们研究结果的一个小样本。我们将不协调集与动力系统中的递归联系起来，在这种情况下，我们展示了不协调集与具有正测度的无处稠密集之间的密切类比。我们对无平方数进行了广泛的推广，在$mathbb{Z}$、$mathbb{Z}^d$和Heisenberg群中产生了许多不协调集的例子。我们开发了一种统一的方法来计算这些不协调集的密度。证明了对于任意可数阿贝尔群$G$， $G$中的任意F{ 0}序列$Phi$，以及$c (0,1)$，存在一个不协调集$ a 子集$G$，且$d_Phi(a) = c$。这里$d_ $表示沿$ $的密度。在此过程中，我们从数学的各个角落，包括经典的拉姆齐理论，遍历理论，数论，拓扑和符号动力学。
{"title":"Discordant sets and ergodic Ramsey theory.","authors":"V. Bergelson, Jake Huryn, R. Raghavan","doi":"10.2140/involve.2022.15.89","DOIUrl":"https://doi.org/10.2140/involve.2022.15.89","url":null,"abstract":"We explore the properties of non-piecewise syndetic sets with positive upper density, which we call discordant, in countable amenable (semi)groups. Sets of this kind are involved in many questions of Ramsey theory and manifest the difference in complexity between the classical van der Waerden's theorem and Szemer'edi's theorem. We generalize and unify old constructions and obtain new results about these historically interesting sets. Here is a small sample of our results. $bullet$ We connect discordant sets to recurrence in dynamical systems, and in this setting we exhibit an intimate analogy between discordant sets and nowhere dense sets having positive measure. $bullet$ We introduce a wide-ranging generalization of the squarefree numbers, producing many examples of discordant sets in $mathbb{Z}$, $mathbb{Z}^d$, and the Heisenberg group. We develop a unified method to compute densities of these discordant sets. $bullet$ We show that, for any countable abelian group $G$, any F{o}lner sequence $Phi$ in $G$, and any $c in (0, 1)$, there exists a discordant set $A subseteq G$ with $d_Phi(A) = c$. Here $d_Phi$ denotes density along $Phi$. Along the way, we draw from various corners of mathematics, including classical Ramsey theory, ergodic theory, number theory, and topological and symbolic dynamics.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75224450","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}
These notes are a written version of my talk given at the CARMA workshop in June 2017, with some additional material. I presented a few concepts that have recently been used in the computation of tree-level scattering amplitudes (mostly using pure spinor methods but not restricted to it) in a context that could be of interest to the combinatorics community. In particular, I focused on the appearance of {it planar binary trees} in scattering amplitudes and presented some curious identities obeyed by related objects, some of which are known to be true only via explicit examples.
{"title":"Planar binary trees in scattering amplitudes","authors":"Carlos R. Mafra","doi":"10.4171/205-1/6","DOIUrl":"https://doi.org/10.4171/205-1/6","url":null,"abstract":"These notes are a written version of my talk given at the CARMA workshop in June 2017, with some additional material. I presented a few concepts that have recently been used in the computation of tree-level scattering amplitudes (mostly using pure spinor methods but not restricted to it) in a context that could be of interest to the combinatorics community. In particular, I focused on the appearance of {it planar binary trees} in scattering amplitudes and presented some curious identities obeyed by related objects, some of which are known to be true only via explicit examples.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88900918","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}
Karola M'esz'aros, Linus Setiabrata, Avery St. Dizier
We give a new operator formula for Grothendieck polynomials that generalizes Magyar's Demazure operator formula for Schubert polynomials. Unlike the usual recursive definition of Grothendieck polynomials, the new formula is ascending in degree. Our proofs are purely combinatorial, contrasting with the geometric and representation theoretic tools used by Magyar for his operator formula for Schubert polynomials. Additionally, our approach yields a new proof of Magyar's formula.
{"title":"An orthodontia formula for Grothendieck polynomials","authors":"Karola M'esz'aros, Linus Setiabrata, Avery St. Dizier","doi":"10.1090/tran/8529","DOIUrl":"https://doi.org/10.1090/tran/8529","url":null,"abstract":"We give a new operator formula for Grothendieck polynomials that generalizes Magyar's Demazure operator formula for Schubert polynomials. Unlike the usual recursive definition of Grothendieck polynomials, the new formula is ascending in degree. Our proofs are purely combinatorial, contrasting with the geometric and representation theoretic tools used by Magyar for his operator formula for Schubert polynomials. Additionally, our approach yields a new proof of Magyar's formula.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85766205","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 : 2020-11-25DOI: 10.2140/involve.2020.13.721
Florence Maas-Gari'epy, Rebecca Patrias
We prove K-theoretic and shifted K-theoretic analogues of the bijection of Stanton and White between domino tableaux and pairs of semistandard tableaux. As a result, we obtain product formulas for pairs of stable Grothendieck polynomials and pairs of K-theoretic Q-Schur functions.
{"title":"Set-valued domino tableaux and shifted set-valued domino tableaux","authors":"Florence Maas-Gari'epy, Rebecca Patrias","doi":"10.2140/involve.2020.13.721","DOIUrl":"https://doi.org/10.2140/involve.2020.13.721","url":null,"abstract":"We prove K-theoretic and shifted K-theoretic analogues of the bijection of Stanton and White between domino tableaux and pairs of semistandard tableaux. As a result, we obtain product formulas for pairs of stable Grothendieck polynomials and pairs of K-theoretic Q-Schur functions.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73810138","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}
Our first main result is a uniform bound, in every dimension $k in mathbb N$, on the topological Turan numbers of $k$-dimensional simplicial complexes: for each $k in mathbb N$, there is a $lambda_k ge k^{-2k^2}$ such that for any $k$-complex $mathcal{S}$, every $k$-complex on $n ge n_0(mathcal{S})$ vertices with at least $n^{k+1 - lambda_k}$ facets contains a homeomorphic copy of $mathcal{S}$. This was previously known only in dimensions one and two, both by highly dimension-specific arguments: the existence of $lambda_1$ is a result of Mader from 1967, and the existence of $lambda_2$ was suggested by Linial in 2006 and recently proved by Keevash-Long-Narayanan-Scott. We deduce this geometric fact from a purely combinatorial result about trace-bounded hypergraphs, where an $r$-partite $r$-graph $H$ with partite classes $V_1, V_2, dots, V_r$ is said to be $d$-trace-bounded if for each $2 le i le r$, all the vertices of $V_i$ have degree at most $d$ in the trace of $H$ on $V_1 cup V_2 cup dots cup V_i$. Our second main result is the following estimate for the Turan numbers of degenerate trace-bounded hypergraphs: for all $r ge 2$ and $dinmathbb N$, there is an $alpha_{r,d} ge (5rd)^{1-r}$ such that for any $d$-trace-bounded $r$-partite $r$-graph $H$, every $r$-graph on $n ge n_0(H)$ vertices with at least $n^{r - alpha_{r,d}}$ edges contains a copy of $H$. This strengthens a result of Conlon-Fox-Sudakov from 2009 who showed that such a bound holds for $r$-partite $r$-graphs $H$ satisfying the stronger hypothesis that the vertex-degrees in all but one of its partite classes are bounded (in $H$, as opposed to in its traces).
我们的第一个主要结果是在每个维度上都有一个统一的界 $k in mathbb N$的拓扑图兰数 $k$-维简单复合体:对于每一个 $k in mathbb N$，有一个。 $lambda_k ge k^{-2k^2}$ 这样对于任何 $k$-复合体 $mathcal{S}$，每 $k$-complex on $n ge n_0(mathcal{S})$ 顶点至少 $n^{k+1 - lambda_k}$ 的同胚副本 $mathcal{S}$． 以前只在维度1和维度2中知道这一点，这两个维度都是高度特定于维度的参数:的存在 $lambda_1$ 是1967年Mader的结果，以及 $lambda_2$ 由Linial在2006年提出，最近由Keevash-Long-Narayanan-Scott证明。我们从一个关于迹有界超图的纯组合结果中推导出这个几何事实 $r$-分 $r$-图 $H$ 有部分类 $V_1, V_2, dots, V_r$ 据说是 $d$-trace-bounded if for each $2 le i le r$的所有顶点 $V_i$ 最多有学位 $d$ 在…的痕迹中 $H$ on $V_1 cup V_2 cup dots cup V_i$． 我们的第二个主要结果是以下对退化迹界超图的图兰数的估计:对于所有 $r ge 2$ 和 $dinmathbb N$，有一个 $alpha_{r,d} ge (5rd)^{1-r}$ 这样对于任何 $d$-trace-bounded $r$-分 $r$-图 $H$，每 $r$-图上 $n ge n_0(H)$ 顶点至少 $n^{r - alpha_{r,d}}$ 的副本 $H$． 这加强了Conlon-Fox-Sudakov在2009年的研究结果，他表明这种界限是成立的 $r$-分 $r$-图 $H$ 满足更强的假设，即它的所有分类的顶点度除了一个以外都是有界的 $H$，而不是in its traces)。
{"title":"Simplicial homeomorphs and trace-bounded hypergraphs","authors":"J. Long, Bhargav P. Narayanan, Corrine Yap","doi":"10.19086/da.36647","DOIUrl":"https://doi.org/10.19086/da.36647","url":null,"abstract":"Our first main result is a uniform bound, in every dimension $k in mathbb N$, on the topological Turan numbers of $k$-dimensional simplicial complexes: for each $k in mathbb N$, there is a $lambda_k ge k^{-2k^2}$ such that for any $k$-complex $mathcal{S}$, every $k$-complex on $n ge n_0(mathcal{S})$ vertices with at least $n^{k+1 - lambda_k}$ facets contains a homeomorphic copy of $mathcal{S}$. This was previously known only in dimensions one and two, both by highly dimension-specific arguments: the existence of $lambda_1$ is a result of Mader from 1967, and the existence of $lambda_2$ was suggested by Linial in 2006 and recently proved by Keevash-Long-Narayanan-Scott. We deduce this geometric fact from a purely combinatorial result about trace-bounded hypergraphs, where an $r$-partite $r$-graph $H$ with partite classes $V_1, V_2, dots, V_r$ is said to be $d$-trace-bounded if for each $2 le i le r$, all the vertices of $V_i$ have degree at most $d$ in the trace of $H$ on $V_1 cup V_2 cup dots cup V_i$. Our second main result is the following estimate for the Turan numbers of degenerate trace-bounded hypergraphs: for all $r ge 2$ and $dinmathbb N$, there is an $alpha_{r,d} ge (5rd)^{1-r}$ such that for any $d$-trace-bounded $r$-partite $r$-graph $H$, every $r$-graph on $n ge n_0(H)$ vertices with at least $n^{r - alpha_{r,d}}$ edges contains a copy of $H$. This strengthens a result of Conlon-Fox-Sudakov from 2009 who showed that such a bound holds for $r$-partite $r$-graphs $H$ satisfying the stronger hypothesis that the vertex-degrees in all but one of its partite classes are bounded (in $H$, as opposed to in its traces).","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77040018","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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