Our work ties together a few different actions studied in combinatorics.  First, we will discuss the action of rowmotion on the set of antichains of a partially-ordered set (poset).  This action, which sends an antichain A to the minimal elements of the complement of the order ideal generated by A, has received significant attention recently in dynamical algebraic combinatorics due to various phenomena (e.g. periodicity, cyclic sieving, homomesy) on certain “nice” posets.  Then, The Lalanne–Kreweras involution (LK) on Dyck paths yields a bijective proof of the symmetry of two statistics: the number of valleys and the major index.  Panyushev studied an equivalent involution that can be considered on the set of antichains of the type A root poset.  The LK involution and rowmotion are connected in that they generate a dihedral action on the set of antichains of the type A root poset.  Furthermore, the periodicity of rowmotion on the type A root poset lifts to a generalization called “birational rowmotion” first studied by David Einstein and James Propp.  This motivated us to search for a birational lifting of the LK involution, where we discovered that the key properties of the LK involution are also satisfied in this generalization. 

This is joint work with Sam Hopkins.

Higher-Weight Dowling Lattices (HWDL in short) are special families of geometric lattices introduced by Dowling in connection with coding theory. The elements of HWDLs are the F_q-linear subspaces of (F_q)ⁿ having a basis of vectors with Hamming weight bounded from above, ordered by inclusion. These lattices were further studied, among others, by Bonin, Kung, and more recently by Ravagnani.

In this talk, we define and investigate structural properties of the q-analogues of HWDLs, which we call rank-metric lattices (RML in short). Their elements are the F_q^m – linear subspaces of (F_q^m)ⁿ having a basis of vectors with rank weight bounded from above, ordered by inclusion. We determine which RMLs are supersolvable, computing their characteristic polynomials. In the second part of the talk, we establish a connection between RMLs and the problem of distinguishing between inequivalent rank-metric codes.

The new results in this talk are joint work with A. Ravagnani

The Gaussian distribution is widely used in mechanism design for differential privacy (DP). Thanks to its sub-Gaussian tail, it significantly reduces the chance of outliers when responding to queries. However, it can only provide approximate DP. In this paper, we introduce and analyze a new distribution for use in DP that is based on the Gaussian distribution, but it has improved privacy performance. The so-called offset-symmetric Gaussian tail (OSGT) distribution is obtained through using the normalized tails of two symmetric Gaussians around zero. Consequently, it can still have sub-Gaussian tail and lend itself to analytical derivations. We analytically derive the variance of the OSGT random variable and its differential privacy metrics. Numerical results show the OSGT mechanism can offer better privacy-utility performance compared to the Gaussian and Laplace mechanisms.