Richard Stanley:The X-Descent Set of a Permutation

Let \(X\)  be a subset of \(\{(i,j) \colon 1\leq i,j \leq n,\ i\neq j\}\). The \(X\)  descent set of a permutation \(w = a_1 \cdots a_n\) of \(1,2,\dots,n\) is defined by

\(\begin{equation} \mathrm{XDes}(w) = \{i \colon (a_i,a_{i+1})\in X\}. \end{equation}\)

If \(X = \{(i,j) \colon n\geq i>j\geq 1\}\), then \(\mathrm{XDes}(w) = \mathrm{Des}(w)\), the ordinary descent set. We define a symmetric function \(U_X\) which is a generating function for permutations \(w\in S_n\) according to their \(X\)-descent set. We will discuss some properties of \(U_X\), including connections with Hamiltonian paths in digraphs. A knowledge of symmetric functions will be helpful but not essential for understanding the talk.