The expected number of cycles of length at most $\mathrm{\ell }$ in $G(n,p)$

Given: Naturals $n,\mathrm{\ell }$ and $p\in [0,1]$.

$G(n,p)$ has mutually independent Bernoulli edge coordinates (The Erdős-Rényi finite random graph $G(n,p)$).

A cycle is a closed walk of length at least $3$ whose vertices, apart from the coinciding endpoints, are distinct (Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges), and the girth is the least length of a cycle (Graph distance within a component, eccentricity, diameter and girth, including the acyclic convention).

A prescribed set of $r$ present edges has probability $p^r$ (A prescribed set of present and absent edges in $G(n,p)$ has product probability).

The falling factorial $n^{\underset{‾}{r}}$ is defined by $n^{\underset{‾}{0}}=1$ and $n^{\underset{‾}{k+1}}=n^{\underset{‾}{k}}(n-k)$ (The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$), and for finite sets $\mathrm{\mid }A\mathrm{\mid }=n$, $\mathrm{\mid }B\mathrm{\mid }=r$ the injections $B\to A$ number $n^{\underset{‾}{r}}$ (The number of injections from a $k$-element set into an $n$-element set is $n^{\underset{‾}{k}}$). An ordered list of $r$ distinct vertices is such an injection, so there are $n^{\underset{‾}{r}}$ of them.

Indicators count occurrences and expectation is linear (Indicators turn event probabilities, intersections, and finite counts into expectations and products, Expectation is linear for every finite family of random variables, without any independence hypothesis).