### Around the Razumovâ€“Stroganov Conjecture: Proof of a Multi-Parameter Sum Rule

#### Abstract

We prove that the sum of entries of the suitably normalized groundstate vector of the $O(1)$ loop model with periodic boundary conditions on a periodic strip of size $2n$ is equal to the total number of $n\times n$ alternating sign matrices. This is done by identifying the state sum of a multi-parameter inhomogeneous version of the $O(1)$ model with the partition function of the inhomogeneous six-vertex model on a $n\times n$ square grid with domain wall boundary conditions.