Some Hecke algebra products and corresponding random walks

Let i=1+q+???+q ^i?1. For certain sequences (r ₁,…,r _l ) of positive integers, we show that in the Hecke algebra ? _n (q) of the symmetric group \(\mathfrak{S}_{n}\), the product \((1+\boldsymbol{r}_{\boldsymbol{1}}T_{r_{1}})\cdots (1+\boldsymbol{r}_{\boldsymbol{l}}T_{r_{l}})\) has a simple explicit expansion in terms of the standard basis {T _w }. An interpretation is given in terms of random walks on \(\mathfrak{S}_{n}\).