In India, to which we most likely owe the decimal place value system for numeration and operations, mathematics developed an art of computation and a science of algorithms to a large extend, probably thanks to an analytical turn of mind best examplified in Panini's grammar. Reading prosody treatises, like Kedāra's, or saṃgīta treatises, like Śārṅgadeva's, we come across certain gems of this art of algorithms. In chapter six of Vṛttaratnākara we find a comprehensive study on how to combine long and short syllables in a metre of a fixed length. In Saṃgītaratnākara, chapter four, we find no less a comprehensive study on how to combine four elementary unit of time in a given measure. All these issues fall in the mathematical field of combinatorics. This paper intends to show that the algorithms used by the Indian paṇḍits to establish all the combinations are an ingenious construction which involves a high knowledge in mathematics: binary expansion of a number in prosody, integer partitions and order structures for those in music. Moreover, these algorithms, by their simple implementation, allow many further developments such as: counting, using recursive formulæ, finding the pattern of a combination by knowing its order amongst the combinations or, conversely, finding the order of a given combination.