Notation for number of descents in a permutation

Question

Let $\sigma \in S_n$ be a permutation. A descent in $\sigma$ is any position $i \in \{1,2,\dots,n-1\}$ for which $\sigma(i+1) < \sigma(i)$.

Is there a standard notation for the number of descents of $\sigma$? That is, $$ \#\{i : \sigma(i+1) < \sigma(i)\}$$

I am reading a paper (by an analyst) in which this is called the number of "errors" of $\sigma$ and denoted $e(\sigma)$, but this notation / terminology does not seem to be standard.

Answer 1

I don't know about any standard notation for this. If you call it descents you may denote the corresponding number by $d(σ)$, which is consistent with $e(σ)$ or $i(σ)$ (see below). Of course you should introduce any notation you will use.

There is also a related notion of inversions of a permutation and the number of inversions is denoted by $\operatorname{inv}(σ)$ or $i(σ)$, see https://en.wikipedia.org/wiki/Inversion_(discrete_mathematics). Your notion corresponds to consecutive inversions.