External invasions in India during the post-Mauryan period had a significant impact on the art and culture of the country. Here are some ways how these invasions influenced Indian art and culture:

1. Introduction of foreign art styles:
External invasions brought new art styles and techniques to India, which influenced the local artists. For example, the Indo-Greek rulers introduced Hellenistic art, which had a significant influence on the development of Indian art.

2. Synthesis of cultures:
The invasions led to a synthesis of cultures, as foreign cultures mixed with the existing Indian culture. This gave rise to a new art form, which was a blend of both the cultures. For example, the Gandhara art, which was a blend of Indian and Greek art.

3. Spread of Buddhism:
The invasions also led to the spread of Buddhism in India. This led to the development of Buddhist art, which had a significant impact on Indian art. The Buddhist art was characterized by the use of symbols, such as the wheel of life, lotus, and Bodhi tree.

4. Islamic influence:
The Islamic invasions in India had a significant impact on Indian art and culture. The Islamic rulers introduced new styles of architecture, such as the use of domes and arches. The Mughal rulers also introduced miniature paintings, which were a blend of Indian and Persian art.

5. Decline of traditional art forms:
The invasions also led to the decline of traditional art forms, such as the Mauryan art. This was due to the fact that the new rulers did not patronize these art forms, and instead, promoted their own art styles.

In conclusion, the external invasions in India during the post-Mauryan period had a significant impact on the art and culture of the country. While some traditional art forms declined, new art forms emerged, which were a blend of different cultures.