Shuffle algebras, lattice paths and quantum toroidal ${\mathfrak{g}\mathfrak{l}}_{n\mathrm{\mid }m}$

We describe and compute various families of commuting elements of the matrix shuffle algebra of type ${\mathfrak{g}\mathfrak{l}}_{n\mathrm{\mid }m}$, which is expected to be isomorphic to quantum toroidal ${\mathfrak{g}\mathfrak{l}}_{n\mathrm{\mid }m}$. Our formulas are given in terms of partial traces of products of $R$-matrices of the quantum affine algebra $U_t({\dot{\mathfrak{g}\mathfrak{l}}}_{n\mathrm{\mid }m})$, and have a lattice path interpretation. Our calculations are based on the machinery of the quantum toroidal algebras and a new anti-homomorphism between matrix shuffle algebras.