Pavel Turek

The wreath conjecture: intervals in Dyck paths, and the wreath matrix

Let $k\leq n$ be two positive integers. In 1974, Baranyai proved that if $k$ divides $n$, then the family of subsets of $[n]$ of size $k$ can be decomposed as a disjoint union of set partitions of $[n]$. While the statement fails to hold if $k$ does not divide $n$, Baranyai conjectured a generalisation of his result for any $k\leq n$ by replacing the partitions of $[n]$ by wreaths, certain families of subsets of $[n]$ of size $k$. This conjecture is now known as the wreath conjecture and remains open for all $k\nmid n$ with a single exception given by $k=2$.
            
The talk will consist of two new approaches to the wreath conjecture. The first one focuses on the case $n=2k+1$. For such $k$ and $n$, we introduce a stronger version of the wreath conjecture involving Dyck paths, and prove a necessary condition for this stronger conjecture to hold by counting the intervals of Dyck paths of a given length and with a given number of falls. The second one is algebraic (and applicable to any $k\leq n$). We rephrase the conjecture using a newly introduced matrix called the wreath matrix and apply results of the representation theory of symmetric groups to analyse its spectrum. This is a joint work with Jan Petr.