Let $w \in \mathrm{Sym}(n)$ for some positive integer $n$. Let $r$ be an involution in $\mathrm{Sym}(n)$, and write it as the product of disjoint transpositions like so: $$r = \prod_{i=1}^k (a_i,b_i) $$ for some positive integer $k$ where $a_i,b_i \in \{1,\ldots,n\}$ and $a_i < b_i$ for each $i=1,\ldots,k$.
I want to know the exact conditions for $w$ and $r$ so that $w <_B wr$ in the Bruhat Order ($<_B$), preferably in terms of inversions.
Below I'll add some information on the Bruhat order.
I expect this has been written down somewhere or is relatively trivial but any help would be greatly appreciated.
The Bruhat Order
Here is a concise characterisation of the Bruhat order on the symmetric group. There is more than one characterisation but this one is from Chapter 2 of Björner and Brenti's 'Combinatorics of Coxeter groups'. Some characterisations may be more useful than others!
For each $i,j = 1,\ldots,n$, define $$w[i,j] := |\{ a \in \{1,\ldots,i\} : w(a) \ge j \}|.$$
Then for all distinct $u,v \in \mathrm{Sym}(n)$, $u <_B v$ if and only if for all $i,j = 1,\ldots,n$, $$u[i,j] \le v[i,j].$$