Counting permutations by alternating runs via Hetyei-Reiner trees

Qiongqiong Pan, Yunze Wang, Jiang Zeng

The generating polynomial of permutations of size $n$, counted by the number of alternating runs, has a root at $-1$ of multiplicity $\lfloor (n-2)/2\rfloor$ for all $n\ge 2$. Th…