  The commutative trigonometric shuffle algebra ${\mathrm A}$ is a space of
symmetric rational functions satisfying certain wheel conditions. We describe a
ring isomorphism between ${\mathrm A}$ and the center of the Hecke algebra
using a realization of the elements of ${\mathrm A}$ as partition functions of
coloured lattice paths associated to the $R$-matrix of $\mathcal
U_{t^{1/2}}(\widehat{gl}_{\infty})$. As an application, we compute under
certain conditions the Hilbert series of the commuting scheme and identify it
with a particular element of the shuffle algebra ${\mathrm A}$, thus providing
a combinatorial formula for it as a "domain wall" type partition function of
coloured lattice paths.
