This paper introduces H-vmunet, a High-order Vision Mamba UNet for medical image segmentation. The proposed model extends the adaptability of 2D-selective-scan (SS2D) by incorporating High-order 2D-selective-scan (H-SS2D) and a High-order visual state space (H-VSS) module. H-SS2D progressively reduces redundant information during SS2D operations through higher-order interactions, while the Local-SS2D module enhances the learning ability of local features. The model is evaluated on three publicly available medical image datasets (ISIC2017, Spleen, and CVC-ClinicDB), demonstrating strong performance in medical image segmentation tasks. The H-vmunet model reduces the number of parameters by 67.28% compared to the traditional Vision Mamba UNet (VM-UNet) model, achieving significant improvements in all three datasets. The model also shows better performance than other state-of-the-art medical image segmentation models, including U-Net, MHorUNet, and VM-UNet. The H-vmunet model is constructed by integrating the H-VSS module into the U-Net framework, enabling higher-order spatial interactions and reducing redundant information. The model's effectiveness is validated through ablation experiments, which show that the H-SS2D module significantly improves segmentation performance. The H-vmunet model is the first to introduce Vision Mamba to higher-order operations for medical image segmentation, demonstrating its potential as a strong contender in the field.This paper introduces H-vmunet, a High-order Vision Mamba UNet for medical image segmentation. The proposed model extends the adaptability of 2D-selective-scan (SS2D) by incorporating High-order 2D-selective-scan (H-SS2D) and a High-order visual state space (H-VSS) module. H-SS2D progressively reduces redundant information during SS2D operations through higher-order interactions, while the Local-SS2D module enhances the learning ability of local features. The model is evaluated on three publicly available medical image datasets (ISIC2017, Spleen, and CVC-ClinicDB), demonstrating strong performance in medical image segmentation tasks. The H-vmunet model reduces the number of parameters by 67.28% compared to the traditional Vision Mamba UNet (VM-UNet) model, achieving significant improvements in all three datasets. The model also shows better performance than other state-of-the-art medical image segmentation models, including U-Net, MHorUNet, and VM-UNet. The H-vmunet model is constructed by integrating the H-VSS module into the U-Net framework, enabling higher-order spatial interactions and reducing redundant information. The model's effectiveness is validated through ablation experiments, which show that the H-SS2D module significantly improves segmentation performance. The H-vmunet model is the first to introduce Vision Mamba to higher-order operations for medical image segmentation, demonstrating its potential as a strong contender in the field.