A Combinatorial Formula for Kazhdan-Lusztig Polynomials of Sparse Paving Matroids

We present a combinatorial formula using skew Young tableaux for the coefficients of Kazhdan-Lusztig polynomials for sparse paving matroids. These matroids are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. We also show the positivity of these coefficients using our formula. In special cases, such as uniform matroids, our formula has a nice combinatorial interpretation.

On the convergence of bumping routes to their limit shapes in the RSK algorithm: numerical experiments

Introduction: The Robinson — Schensted — Knuth (RSK) algorithm transforms a sequence of elements of a linearly ordered set into a pair of Young tableaux P, Q of the same shape. This transformation is based on the process of bumping and inserting elements in tableau P according to special rules. The trajectory formed by all the boxes of the tableau P shifted in the RSK algorithm is called the bumping route. D. Romik and P. Śniady in 2016 obtained an explicit formula for the limit shape of the bumping route, which is determined by its first element. However, the rate of convergence of the bumping routes to the limit shape has not been previously investigated either theoretically or by numerical experiments. Purpose: Carrying out computer experiments to study the dynamics of the bumping routes produced by the RSK algorithm on Young tableaux as their sizes increase. Calculation of statistical means and variances of deviations of bumping routes from their limit shapes in the L2 metric for various values fed to the input of the RSK algorithm. Results: A series of computer experiments have been carried out on Young tableaux, consisting of up to 10 million boxes. We used 300 tableaux of each size. Different input values (0.1, 0.3, 0.5, 0.7, 0.9) were added to each such tableau using the RSK algorithm, and the deviations of the bumping routes built from these values from the corresponding limit shapes were calculated. The graphs of the statistical mean values and variances of these deviations were produced. It is noticed that the deviations decrease in proportion to the fourth root of the tableau size n. An approximation of the dependence of the mean values of deviations on n was obtained using the least squares method.

Asymptotics for the number of standard tableaux of skew shape and for weighted lozenge tilings

We prove and generalise a conjecture in [MPP4] about the asymptotics of $\frac{1}{\sqrt{n!}} f^{\lambda/\mu}$ , where $f^{\lambda/\mu}$ is the number of standard Young tableaux of skew shape $\lambda/\mu$ which have stable limit shape under the $1/\sqrt{n}$ scaling. The proof is based on the variational principle on the partition function of certain weighted lozenge tilings.

Symplectic Keys and Demazure Atoms in Type C

We compute, mimicking the Lascoux-Schützenberger type A combinatorial procedure, left and right keys for a Kashiwara-Nakashima tableau in type C. These symplectic keys have a similar role as the keys for semistandard Young tableaux. More precisely, our symplectic keys give a tableau criterion for the Bruhat order on the hyperoctahedral group and cosets, and describe Demazure atoms and characters in type C. The right and the left symplectic keys are related through the Lusztig involution. A type C Schützenberger evacuation is defined to realize that involution.

Row bounds needed to justifiably express flagged Schur functions with Gessel-Viennot determinants

Let $\lambda$ be a partition with no more than $n$ parts. Let $\beta$ be a weakly increasing $n$-tuple with entries from $\{ 1, ... , n \}$. The flagged Schur function in the variables $x_1, ... , x_n$ that is indexed by $\lambda$ and $\beta$ has been defined to be the sum of the content weight monomials for the semistandard Young tableaux of shape $\lambda$ whose values are row-wise bounded by the entries of $\beta$. Gessel and Viennot gave a determinant expression for the flagged Schur function indexed by $\lambda$ and $\beta$; this could be done since the pair $(\lambda, \beta)$ satisfied their "nonpermutable" condition for the sequence of terminals of an $n$-tuple of lattice paths that they used to model the tableaux. We generalize flagged Schur functions by dropping the requirement that $\beta$ be weakly increasing. Then for each $\lambda$ we give a condition on the entries of $\beta$ for the pair $(\lambda, \beta)$ to be nonpermutable that is both necessary and sufficient. When the parts of $\lambda$ are not distinct there will be multiple row bound $n$-tuples $\beta$ that will produce the same set of tableaux. We accordingly group the bounding $\beta$ into equivalence classes and identify the most efficient $\beta$ in each class for the determinant computation. We recently showed that many other sets of objects that are indexed by $n$ and $\lambda$ are enumerated by the number of these efficient $n$-tuples. We called these counts "parabolic Catalan numbers". It is noted that the $GL(n)$ Demazure characters (key polynomials) indexed by 312-avoiding permutations can also be expressed with these determinants. Comment: 22 pages, 5 figures, 4 tables. Identical to v.5, except for the insertion of a reference and the DMTCS journal's publication meta data

A maxdrop statistic for standard Young tableaux

In this paper, we introduce a new statistic on standard Young tableaux that is closely related to the maxdrop permutation statistic that was introduced by the first author. We prove that the value of the statistic must be attained at one of the corners of the standard Young tableau. We determine the coefficients of the generating function of this statistic over two-row standard Young tableaux having [Formula: see text] cells. We prove several results for this new statistic that include unimodality of the coefficients for the two-row case.