Enumerations of half-turn-symmetric alternating-sign matrices of odd order

Kuperberg showed that the partition function of the square-ice model related to half-turn-symmetric alternating-sign matrices of even order is the product of two similar factors. We propose a square-ice model whose states are in bijective correspondence with half-turn-symmetric alternating-sign matrices of odd order. The partition function of this model is expressed via the above factors. We find the contributions to the partition function that correspond to the alternating-sign matrices having $1$ or $-1$ as the central entry and establish the related enumerations.