Matrix product operator representations for the local conserved quantities of the spin-$1/2$ XYZ chain

We present explicit matrix product operator (MPO) representations for the local conserved quantities of the spin-$1/2$ XYZ chain. Through these MPO representations, we simplify the coefficients appearing in the local conserved quantities originally derived by one of the authors, and reveal their combinatorial meaning: the coefficients prove to be a polynomial generalization of the Catalan numbers, defined via weighted monotonic lattice paths. Furthermore, we obtain a new simple $3 \times 3$ Lax operator for the XYZ chain that, unlike Baxter's R-matrix, does not involve elliptic functions.