Editing Pingala (section)

==Combinatorics==
The ''{{IAST|Chandaḥśāstra}}'' presents a formula to generate systematic enumerations of [[Metre (poetry)|metres]], of all possible combinations of [[Sanskrit prosody#Light and heavy syllables|light (''laghu'') and heavy (''guru'') syllables]], for a word of ''n'' syllables, using a recursive formula, that results in a partially ordered [[binary numeral system|binary]] representation.<ref>Van Nooten (1993)</ref> Pingala is credited with being the first to express the [[combinatorics]] of [[Sanskrit prosody|Sanskrit metre]], eg.<ref>{{Cite journal |last=Hall |first=Rachel Wells |date=February 2008 |title=Math for Poets and Drummers |url=https://www.jstor.org/stable/25678735 |journal=Math Horizons |publisher=[[Taylor & Francis]] |volume=15 |issue=3 |pages=10{{en dash}}12 |doi=10.1080/10724117.2008.11974752 |jstor=25678735 |s2cid=3637061 |access-date=27 May 2022 }}</ref>
* Create a syllable list ''x'' comprising one light (''L'') and heavy (''G'') syllable:
* Repeat till list ''x'' contains only words of the desired length ''n''
** Replicate list ''x'' as lists ''a'' and ''b''
*** Append syllable ''L'' to each element of list ''a''
*** Append syllable ''G'' to each element of list ''b''
** Append lists ''b'' to list ''a'' and rename as list ''x''

{| class="wikitable"
|+ Possible combinations of ''Guru'' and ''Laghu'' syllables in a word of length ''n''<ref>{{Cite web |last=Shah |first=Jayant |title=A History of Pingala's Combinatorics  |url=https://web.northeastern.edu/shah/papers/Pingala.pdf}}</ref>
|-
!Word length (''n'' characters)!!Possible combinations
|-
| 1 || G L
|-
| 2 || GG LG GL LL
|-
| 3 || GGG LGG GLG LLG GGL LGL GLL LLL
|-
|}

Because of this, Pingala is sometimes also credited with the first use of [[0|zero]], as he used the [[Sanskrit]] word ''[[Śūnyatā|śūnya]]'' to explicitly refer to the number.<ref>{{harvtxt|Plofker|2009}}, pp. 54–56: "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, [...] Pingala's use of a zero symbol [śūnya] as a marker seems to be the first known explicit reference to zero. ... In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, there are five questions concerning the possible meters for any value "n". [...] The answer is (2)<sup>7</sup> = 128, as expected, but instead of seven doublings, the process (explained by the sutra) required only three doublings and two squarings – a handy time saver where "n" is large. Pingala's use of a zero symbol as a marker seems to be the first known explicit reference to zero."</ref> Pingala's binary representation increases towards the right, and not to the left as modern [[binary numbers]] usually do.<ref>{{Cite book|title=The mathematics of harmony: from Euclid to contemporary mathematics and computer science|first1=Alexey|last1=Stakhov|author1-link=Alexey Stakhov|first2=Scott Anthony|last2=Olsen|isbn=978-981-277-582-5|year=2009|publisher=World Scientific |url=https://books.google.com/books?id=K6fac9RxXREC}}</ref> In Pingala's system, the numbers start from number one, and not zero. 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 value]]s.<ref>B. van Nooten, "Binary Numbers in Indian Antiquity", Journal of Indian Studies, Volume 21, 1993, pp. 31–50</ref> Pingala's work also includes material related to the [[Fibonacci numbers]], called ''{{IAST|mātrāmeru}}''.<ref>{{cite book |title = Toward a Global Science | author = Susantha Goonatilake |publisher = Indiana University Press |year = 1998 |page = [https://archive.org/details/towardglobalscie0000goon/page/126 126] |isbn = 978-0-253-33388-9 |url = https://archive.org/details/towardglobalscie0000goon |url-access = registration |quote = Virahanka Fibonacci. }}</ref>