Metrics and combinatorics: naṣṭaḥ or “lost”

We saw that one way to determine all of the possible combinations of n items taken k at a time is to list them according to a certain procedure, called prastāraḥ or “spreading out.” This procedure assigns a serial number to each of the possible combinations. Now supposing you don’t want to do the entire prastāraḥ, but simply want to figure out the combination that occurs at a given serial number. For instance, among the combinations of 4 syllables, each of which can be either light (।) or heavy (ऽ), what is the exact combination that occurs at serial number 13 in the prastāraḥ?

The procedure for finding out this “lost” (naṣṭaḥ) pattern is as follows:

  1. Start with the serial number of the lost pattern (r).
  2. For each position in the pattern:
    1. if the number is even, the syllable in the position is light; divide this number by 2, and return to step 2;
    2. if the number is odd, the syllable in the position is heavy; add one to this number, then divide the sum by two, and return to step 2.

Hence for r = 13 in the prastāraḥ of four positions (k = 4):

  • First syllable: 13 is odd, hence heavy (ऽ).
  • Second syllable: (13+1)/2 = 7 is odd, hence heavy (ऽ).
  • Third syllable: (7+1)/2 = 4 is even, hence light (।).
  • Fourth syllable: 4/2 = 2 is even, hence light (।).

Hence the lost pattern is ऽऽ।।, which can be verified in the prastāraḥ given as an example in the previous post.