Acharya Pingala

Combinatorics The presents a recursive method to systematically enumerate metres by generating all possible combinations of light (laghu) and heavy (guru) syllables for a verse of syllables, producing a binary representation. Pingala systematized Sanskrit metrics using six algorithmic procedures known as Pratyayas:

Prastāra (permutation/expansion): A systematic, recursive table generating all combinations of laghu (light, short) and guru (heavy, long) syllables for an -syllable meter. The two syllable types can be represented as binary symbols, so the procedure corresponds mathematically to enumerating all binary sequences. Naṣṭa (Recovery): An algorithm to determine the exact syllable sequence, given the rank/index number of a meter. Uddiṣṭa (Indexing): The inverse algorithm to find the rank/index number of a given sequence of syllabus. Laghu-kriyā/Guru-kriyā (Weight determination): Computing the number of metres containing a specific count of light or heavy syllables (yielding binomial coefficients). Saṅkhyā (Total calculation): Determining the total number of permutations for a meter of length . Mātrā-meru/Meru-prastāra (Pyramidal arrangement): The triangular array of combinatorial binomial coefficients (later known in Europe as Pascal's triangle) and additive sequences (later known as Fibonacci numbers). + Metrical combinations generated via Prastāra for length His binary system increases from left to right, rather than right to left as in modern binary notation. Four short syllables "0000" is the first pattern and corresponds to the value one. The numerical value is obtained by adding one to the sum of place values. The combinatorial rules for metres developed by Pingala that were based on moraic time units (mātrās) were the mathematical foundation for the sequence known as Fibonacci numbers in the West. This sequence of numbers was later formalized by Indian mathematicians Virahānka (c. 6th–8th century CE) and Hemachandra (c. 1150 CE), centuries prior to Fibonacci.

Editions A. Weber, Indische Studien 8, Leipzig, 1863. Janakinath Kabyatittha & Brothers, Pingala Chhanda Sutram, Calcutta, 1931. Nirnayasagar Press, Chand Shastra, Bombay, 1938.