## Combinatorics and Graph Theory

•  Sergi Elizalde, Dartmouth College
•  Descents on quasi-Stirling permutations
•  01/20/2021
•  3:00 PM - 3:50 PM
•  Online (virtual meeting) (Virtual Meeting Link)
•  Bruce E Sagan (bsagan@msu.edu)

Stirling permutations were introduced by Gessel and Stanley to give a combinatorial interpretation of certain polynomials related to Stirling numbers. A very natural extension of Stirling permutations are quasi-Stirling permutations, which are in bijection with labeled rooted plane trees. Archer et al. introduced these permutations, and conjectured that there are $(n+1)^{n-1}$ quasi-Stirling permutations of size $n$ having $n$ descents. In this talk we prove this conjecture. More generally, we give the generating function for quasi-Stirling permutations by the number of descents, which turns out to satisfy a beautiful equation involving Eulerian polynomials. We show that some of the properties of descents on usual permutations and on Stirling permutations have an analogue for quasi-Stirling permutations. Finally, we extend our results to a one-parameter family of permutations, called $k$-quasi-Stirling permutations, which are in bijection with certain decorated trees.

## Contact

Department of Mathematics
Michigan State University