{"title":"Sorting probability for large Young diagrams","authors":"Swee Hong Chan, I. Pak, G. Panova","doi":"10.19086/da.30071","DOIUrl":"https://doi.org/10.19086/da.30071","url":null,"abstract":"For a finite poset $P=(X,prec)$, let $mathcal{L}_P$ denote the set of linear extensions of $P$. The sorting probability $delta(P)$ is defined as \u0000[delta(P) , := , min_{x,yin X} , bigl| mathbf{P} , [L(x)leq L(y) ] - mathbf{P} , [L(y)leq L(x) ] bigr|,, ] where $L in mathcal{L}_P$ is a uniform linear extension of $P$. We give asymptotic upper bounds on sorting probabilities for posets associated with large Young diagrams and large skew Young diagrams, with bounded number of rows.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78891272","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 absolute points of correlations in PG$(2,q^n)$","authors":"J. D'haeseleer, N. Durante","doi":"10.37236/abcd","DOIUrl":"https://doi.org/10.37236/abcd","url":null,"abstract":"Let $V$ be a $(d+1)$-dimensional vector space over a field $mathbb{F}$. Sesquilinear forms over $V$ have been largely studied when they are reflexive and hence give rise to a (possibly degenerate) polarity of the $d$-dimensional projective space PG$(V)$. Everything is known in this case for both degenerate and non-degenerate reflexive forms if $mathbb{F}$ is either ${mathbb{R}}$, ${mathbb{C}}$ or a finite field ${mathbb{F}}_q$. In this paper we consider degenerate, non-reflexive sesquilinear forms of $V=mathbb{F}_{q^n}^3$. We will see that these forms give rise to degenerate correlations of PG$(2,q^n)$ whose set of absolute points are, besides cones, the (possibly degenerate) $C_F^m$-sets. In the final section we collect some results from the huge work of B.C. Kestenband regarding what is known for the set of the absolute points of correlations in PG$(2,q^n)$ induced by a non-degenerate, non-reflexive sesquilinear form of $V=mathbb{F}_{q^n}^3$.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74049247","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":"Cayley Graphs Without a Bounded Eigenbasis","authors":"A. Sah, Mehtaab Sawhney, Yufei Zhao","doi":"10.1093/imrn/rnaa298","DOIUrl":"https://doi.org/10.1093/imrn/rnaa298","url":null,"abstract":"Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. \u0000On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(sqrt{log n / n})$, and that this bound is nearly best possible. \u0000Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon--Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor's result to nonabelian groups.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87528499","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":"A short proof of the canonical polynomial van der Waerden theorem","authors":"J. Fox, Yuval Wigderson, Yufei Zhao","doi":"10.5802/crmath.101","DOIUrl":"https://doi.org/10.5802/crmath.101","url":null,"abstract":"We present a short new proof of the canonical polynomial van der Waerden theorem, recently established by Girao [arXiv:2004.07766].","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74089492","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":"Recent progress in combinatorial random matrix theory","authors":"V. Vu","doi":"10.1214/20-PS346","DOIUrl":"https://doi.org/10.1214/20-PS346","url":null,"abstract":"We discuss recent progress many problems in random matrix theory of a combinatorial nature, including several breakthroughs that solve long standing famous conjectures.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72589129","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 families of highly neighborly centrally symmetric spheres","authors":"I. Novik, Hailun Zheng","doi":"10.1090/tran/8631","DOIUrl":"https://doi.org/10.1090/tran/8631","url":null,"abstract":"In 1995, Josckusch constructed an infinite family of centrally symmetric (cs, for short) triangulations of $3$-spheres that are cs-$2$-neighborly. Recently, Novik and Zheng extended Jockusch's construction: for all $d$ and $n>d$, they constructed a cs triangulation of a $d$-sphere with $2n$ vertices, $Delta^d_n$, that is cs-$lceil d/2rceil$-neighborly. Here, several new cs constructions are provided. It is shown that for all $k>2$ and a sufficiently large $n$, there is another cs triangulation of a $(2k-1)$-sphere with $2n$ vertices that is cs-$k$-neighborly, while for $k=2$ there are $Omega(2^n)$ such pairwise non-isomorphic triangulations. It is also shown that for all $k>2$ and a sufficiently large $n$, there are $Omega(2^n)$ pairwise non-isomorphic cs triangulations of a $(2k-1)$-sphere with $2n$ vertices that are cs-$(k-1)$-neighborly. The constructions are based on studying facets of $Delta^d_n$, and, in particular, on some necessary and some sufficient conditions similar in spirit to Gale's evenness condition. Along the way, it is proved that Jockusch's spheres $Delta^3_n$ are shellable and an affirmative answer to Murai-Nevo's question about $2$-stacked shellable balls is given.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81284863","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":"Petrie symmetric functions","authors":"Darij Grinberg","doi":"10.5802/alco.232","DOIUrl":"https://doi.org/10.5802/alco.232","url":null,"abstract":"For any positive integer $k$ and nonnegative integer $m$, we consider the symmetric function $Gleft( k,mright)$ defined as the sum of all monomials of degree $m$ that involve only exponents smaller than $k$. We call $Gleft( k,mright)$ a \"Petrie symmetric function\" in honor of Flinders Petrie, as the coefficients in its expansion in the Schur basis are determinants of Petrie matrices (and thus belong to $left{ 0,1,-1right} $ by a classical result of Gordon and Wilkinson). More generally, we prove a Pieri-like rule for expanding a product of the form $Gleft( k,mright) cdot s_{mu}$ in the Schur basis whenever $mu$ is a partition; all coefficients in this expansion belong to $left{ 0,1,-1right} $. We also show that $Gleft( k,1right) ,Gleft( k,2right) ,Gleft( k,3right) ,ldots$ form an algebraically independent generating set for the symmetric functions when $1-k$ is invertible in the base ring, and we prove a conjecture of Liu and Polo about the expansion of $Gleft( k,2k-1right)$ in the Schur basis.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"118 24","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141210484","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":"Higher connectivity of the Morse complex","authors":"N. Scoville, Matthew C. B. Zaremsky","doi":"10.1090/bproc/115","DOIUrl":"https://doi.org/10.1090/bproc/115","url":null,"abstract":"The Morse complex $mathcal{M}(Delta)$ of a finite simplicial complex $Delta$ is the complex of all gradient vector fields on $Delta$. In particular $mathcal{M}(Delta)$ encodes all possible discrete Morse functions (in the sense of Forman) on $Delta$. In this paper we find sufficient conditions for $mathcal{M}(Delta)$ to be connected or simply connected, in terms of certain measurements on $Delta$. When $Delta=Gamma$ is a graph we get similar sufficient conditions for $mathcal{M}(Gamma)$ to be $(m-1)$-connected. The main technique we use is Bestvina-Brady discrete Morse theory, applied to a \"generalized Morse complex\" that is easier to analyze.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74963889","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":"Patterns without a popular difference","authors":"A. Sah, Mehtaab Sawhney, Yufei Zhao","doi":"10.19086/da.25317","DOIUrl":"https://doi.org/10.19086/da.25317","url":null,"abstract":"Which finite sets $P subseteq mathbb{Z}^r$ with $|P| ge 3$ have the following property: for every $A subseteq [N]^r$, there is some nonzero integer $d$ such that $A$ contains $(alpha^{|P|} - o(1))N^r$ translates of $d cdot P = {d p : p in P}$, where $alpha = |A|/N^r$? \u0000Green showed that all 3-point $P subseteq mathbb{Z}$ have the above property. Green and Tao showed that 4-point sets of the form $P = {a, a+b, a+c, a+b+c} subseteq mathbb{Z}$ also have the property. We show that no other sets have the above property. Furthermore, for various $P$, we provide new upper bounds on the number of translates of $d cdot P$ that one can guarantee to find.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90244061","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":"Symmetric Decompositions and the Veronese Construction","authors":"Katharina Jochemko","doi":"10.1093/IMRN/RNAB031","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB031","url":null,"abstract":"We study rational generating functions of sequences ${a_n}_{ngeq 0}$ that agree with a polynomial and investigate symmetric decompositions of the numerator polynomial for subsequences ${a_{rn}}_{ngeq 0}$. We prove that if the numerator polynomial for ${a_n}_{ngeq 0}$ is of degree $s$ and its coefficients satisfy a set of natural linear inequalities then the symmetric decomposition of the numerator for ${a_{rn}}_{ngeq 0}$ is real-rooted whenever $rgeq max {s,d+1-s}$. Moreover, if the numerator polynomial for ${a_n}_{ngeq 0}$ is symmetric then we show that the symmetric decomposition for ${a_{rn}}_{ngeq 0}$ is interlacing. We apply our results to Ehrhart series of lattice polytopes. In particular, we obtain that the $h^ast$-polynomial of every dilation of a $d$-dimensional lattice polytope of degree $s$ has a real-rooted symmetric decomposition whenever the dilation factor $r$ satisfies $rgeq max {s,d+1-s}$. If the polytope is Gorenstein then this decomposition is moreover interlacing.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74867625","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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