Tags: #definition

statistic on permutations

A statistic on $S_n$ is any map $\sigma :S_n\to {\mathbb{Z}}_{\ge 0}$.

In particular, we often care about the different statistics modulo equidistribution.

Examples

Three important classes of statistics:
Example: Let $w=6,2,4,1,10,3,5,9,8,7$

• Eulerian statistic - the number of descent of a permutation
  • $\mathrm{desc}(w)=5$
• Mahonian statistic - the number of inversion of a permutation
  • $\mathrm{inv}(w)=16$
• Sterlingian statistic - the number of cycles of a permutation
  • $\mathrm{cyc}(w)=4$

Properties