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Contributions of Some Jain Ācāryas to Combinatorics
93
and (for 11 being odd) S is written. Now, since 6 is even, it is halved; 3 is obtained and (for 6 being even) I is written. Again, 3 being odd, I is added to it and the sum is halved. Thus, 2 is obtained and (for 3 being odd) S is written. Now, since 2 is even, it is halved; 1 is ob:ained and (for 2 being even) I is written. Since the metre has 4 letters and 4 letters have already been obtained, the process is stopped at this stage and the leiters, above obtained and written in order, i.e., SIS), in this case, give the desired 11th variation of the metre.
Rule for Uddiştāńka.
The uddistu analysis is the process of knowing the position (or the serial number, so to say) of a given variation (vrtta) amidst ail variations of the metre.
Ācāraya Jayadeva's rule for the same may be translated as given below.
"Unity, each time doubled, (i.e., 1, 2, 4, 8,...) is written over the letters of the given variation, in order. Sum of numbers above mono. syllabic letters, added to unity, give the serial number of the given variation".12
Let us find the serial number of the variation SISI amidst all variations of the metre baviog 4 letters. Numbers 1, 2, 4 and
12 4 8 8 are written over the given variation, in order, thus : SISI. Unity is added to the sum of numbe is above monosyllabic letters. Thus, we obtain, (2+8)+1=-11. So, 11 is the serial number of the given variation amidst all variations of the metre havive 4 letters.
Rule for Sankhyā.
Acārya Jayadeva also considers an enumerative problem of combinatorics in metric viz., sankhyä analysis. Sankhyā analysis is the knowledge of the process of finding the number of variations of a metre without going through the expansion of the metre.
Ācārya Jayadeva's rule for finding the number of variations of a given metre may be translated as given below.
"Sum of (all) the numbers (written above the letters) in finding the uddistānka, added to unity, gives the number of variations of the metre". 18
Thus, the number of variations of a metre having n letters (1+2+4+...+2n-1) + 1.
12. Ja., viii, 6-7. 13. Ja., viii, 11 (b). The text is,
उद्दिष्टविधानाङ्कः सैकमिश्री भवन्त्यथवा ॥ ॥
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