The correct chronological order in which the four personalities appeared in India is as follows: 4, 3, 2, 1.

Panini (4) is believed to have lived around the 6th to 4th century BCE. He was a Sanskrit grammarian and is famous for his work on Sanskrit grammar, known as Ashtadhyayi.

Kautilya (also known as Chanakya) (3) was a philosopher, economist, and advisor to the Mauryan emperor Chandragupta. He is best known for his political treatise, the Arthashastra. He is estimated to have lived from the 4th century BCE to the 3rd century BCE.

Ashwaghosa (2) was a Buddhist poet and philosopher who lived in the 1st century CE. He is known for his works like Buddhacharita and Saundarananda.

Varahamihira (1) was an ancient Indian astronomer, mathematician, and astrologer. He lived between the 5th and 6th century CE and is known for his work on astronomy, the Brihat-Samhita.

Therefore, the correct chronological order is 4, 3, 2