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Register a new userOn a rank-unimodality conjecture of Morier-Genoud and Ovsienko
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Let alpha = (a,b,...) be a composition. Consider the associated poset F(alpha), called a fence, whose covering relations are x_1 < x_2 < ... < x_{a+1} > x_{a+2} > ... > x_{a+b+1} < x_{a+b+2} < ... . We study the associated distributive lattice L(alpha) consisting of all lower order ideals of F(alpha). These lattices are important in the theory of cluster algebras and their rank generating functions can be used to define q-analogues of rational numbers. In particular, we make progress on a recent conjecture of Morier-Genoud and Ovsienko that L(alpha) is rank unimodal. We show that if one of the parts of alpha is greater than the sum of the others, then the conjecture is true. We conjecture that L(alpha) enjoys the stronger properties of having a nested chain decomposition and having a rank sequence which is either top or bottom interlacing, the latter being a recently defined property of sequences. We verify that these properties hold for compositions with at most three parts and for what we call d-divided posets, generalizing work of Claussen and simplifying a construction of Gansner.
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In the study of Kostka numbers and Catalan numbers, Kirillov posed a unimodality conjecture for the rectangular Narayana polynomials. We prove that the rectangular Narayana polynomials have only real zeros, and thereby confirm Kirillovs unimodality conjecture with the help of Newtons inequality. By using an equidistribution property between descent numbers and ascent numbers on ballot paths due to Sulanke and a bijection between lattice words and standard Young tableaux, we show that the rectangular Narayana polynomial is equal to the descent generating function on standard Young tableaux of certain rectangular shape, up to a power of the indeterminate. Then we obtain the real-rootedness of the rectangular Narayana polynomial based on Brentis result that the descent generating function of standard Young tableaux has only real zeros.
A polynomial $A(q)=sum_{i=0}^n a_iq^i$ is said to be unimodal if $a_0le a_1le cdots le a_kge a_{k+1} ge cdots ge a_n$. We investigate the unimodality of rational $q$-Catalan polynomials, which is defined to be $C_{m,n}(q)= frac{1}{[n+m]} left[ m+n atop nright]$ for a coprime pair of positive integers $(m,n)$. We conjecture that they are unimodal with respect to parity, or equivalently, $(1+q)C_{m+n}(q)$ is unimodal. By using generating functions and the constant term method, we verify our conjecture for $mle 5$ in a straightforward way.
We study the log-rank conjecture from the perspective of point-hyperplane incidence geometry. We formulate the following conjecture: Given a point set in $mathbb{R}^d$ that is covered by constant-sized sets of parallel hyperplanes, there exists an affine subspace that accounts for a large (i.e., $2^{-{operatorname{polylog}(d)}}$) fraction of the incidences. Alternatively, our conjecture may be interpreted linear-algebraically as follows: Any rank-$d$ matrix containing at most $O(1)$ distinct entries in each column contains a submatrix of fractional size $2^{-{operatorname{polylog}(d)}}$, in which each column contains one distinct entry. We prove that our conjecture is equivalent to the log-rank conjecture. Motivated by the connections above, we revisit well-studied questions in point-hyperplane incidence geometry without structural assumptions (i.e., the existence of partitions). We give an elementary argument for the existence of complete bipartite subgraphs of density $Omega(epsilon^{2d}/d)$ in any $d$-dimensional configuration with incidence density $epsilon$. We also improve an upper-bound construction of Apfelbaum and Sharir (SIAM J. Discrete Math. 07), yielding a configuration whose complete bipartite subgraphs are exponentially small and whose incidence density is $Omega(1/sqrt d)$. Finally, we discuss various constructions (due to others) which yield configurations with incidence density $Omega(1)$ and bipartite subgraph density $2^{-Omega(sqrt d)}$. Our framework and results may help shed light on the difficulty of improving Lovetts $tilde{O}(sqrt{operatorname{rank}(f)})$ bound (J. ACM 16) for the log-rank conjecture; in particular, any improvement on this bound would imply the first bipartite subgraph size bounds for parallel $3$-partitioned configurations which beat our generic bounds for unstructured configurations.
Motivated by a hat guessing problem proposed by Iwasawa cite{Iwasawa10}, Butler and Graham cite{Butler11} made the following conjecture on the existence of certain way of marking the {em coordinate lines} in $[k]^n$: there exists a way to mark one point on each {em coordinate line} in $[k]^n$, so that every point in $[k]^n$ is marked exactly $a$ or $b$ times as long as the parameters $(a,b,n,k)$ satisfies that there are non-negative integers $s$ and $t$ such that $s+t = k^n$ and $as+bt = nk^{n-1}$. In this paper we prove this conjecture for any prime number $k$. Moreover, we prove the conjecture for the case when $a=0$ for general $k$.
Karasev conjectured that for any set of $3k$ lines in general position in the plane, which is partitioned into $3$ color classes of equal size $k$, the set can be partitioned into $k$ colorful 3-subsets such that all the triangles formed by the subsets have a point in common. Although the general conjecture is false, we show that Karasevs conjecture is true for lines in convex position. We also discuss possible generalizations of this result.
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