03 combinatorics

8/10/2019 03 Combinatorics

1/38

MATHEMATICS AND OTHER STRAS:

DEVELOPMENT OF COMBINATORICS

M.D.SRINIVAS

CENTRE FOR POLICY STUDIES

[email protected]


2/38

COMBINATORICS IN YURVEDA

The ancient Indian medical treatises of Caraka and Suruta (prior to 500

BCE) deal with certain combinatoric questions in relation to the six Rasasand the three Doas. For instance, Caraka (Strasthana Ch. 26) discusses

the 63 combinations that are possible from the 6 Rasas:

...


3/38

COMBINATORICS IN YURVEDA

The number of Bheda or combinations that can be obtained by combining

different number of Rasas is 63 and that is obtained as follows:The number of combinations of 2 rasas selected from the 6 is 15

The number of combinations of 3 rasas selected from the 6 is 20





Hence the total number of Bhedas is 63

This is a particular case of the relation

nC1+nC2

+ ... +nCn = 2n- 1

The Suruta-sahit (Uttarasthna Ch. 63) actually lists each of thesepossibilities in a sequential enumeration akin to a prastra.


4/38

GANDHAYUKTI OF VARHAMIHIRA

Chapter 76 of the great compilation Bhatsahitof Varhamihira (c.550)

is devoted to a discussion of perfumery. In verse 20, Varha mentions thatthere are 1,820 combinations which can be formed by choosing 4

perfumes from a set of 16 basic perfumes (16C4= 1820).

In verse 22, Varha a method of construction of a Meru (or a tabular

figure) which may be used to calculate the number of combinations. This

verse also very briefly indicates a way of arranging these combinations in

an array or a prastra.

: :

Bhaotpala (c.950) in his commentary has explained both the constructionof the Meru and the method of loa-prastra of the combinations.


5/38

GANDHAYUKTI OF VARHAMIHIRA

In the first column the natural numbers are written. In the second column,

their sums, in the third the sums of sums, and so on. One row is reduced ateach step. The top entry in each column gives the number ofcombinations. The above Meru is based on the relation

nCr=n-1Cr-1 +

n-2Cr-1 +.......+r-1Cr-1

16

15 12014 105 56013 91 455 182012 78 364 136511 66 286 100110 55 220 715

9 45 165 4958 36 120 3307 28 84 2106 21 56 1265 15 35 704 10 20 353 6 10 152 3 4 51 1 1 1


6/38

MTR-VTTA

By assigning the values 1 for Laghu and 2 for Guru, we can obtain the

total value or mtrs, associated with each metrical form. This leads to thenotion of mtr-vtttas or moric metres, where the metrical patterns are

classified by their total value or mtr.

Pigala has only briefly touched upon mtr-vttas in Chapter IV of

Chanda-strawhile discussing the various forms of ryand Vaitlyavttas.

Mtr-vttas are more commonly met with in Prkta and regionallanguages. The Prkta work Vttajti-samuccaya of Virahka (c.600)

discusses the pratyayas of prastra and sakhy

for m

tr-vttas. A moredetailed discussion of mtrvttas is available in Chandonusana of

Hemacandra (c.1200), Prkta-Paigala, Vbhaa of Dmodara

(c.1500) and the commentary of Nryaabhaa (c.1550) onVttaratnkaraof Kedra (c.1000)


7/38

MTR-PRASTRA

( .)

The prastra in the case of a mtr-vtta of n-mtrs is to be generated

following the same procedure as in the case of a vara-vtta except for thefollowing:

The first row consists of all Gs if n is even and an L followed by

all Gs if nis odd.

[Given any row in the prastra, to generate the next line, scan from

the left to identify the first G. Place an L below that. The elementsto the right are brought down as they are.]

The remaining mtrs to the left are filled in by all Gs, and byplacing an L at the beginning if need be to keep the number of

mtrs the same.


8/38

MTR-PRASTRA

1-MtrPrastra

1 G

2-MtrPrastra

1 G2 L L

3-MtrPrastra

1 L G2 G L

3 L L L4-MtrPrastra

1 G G2 L L G

3 L G L4 G L L5 L L L L


9/38

MTR-PRASTRA

6-MtrPrastra1 G G G2 L L G G3 L G L G4 G L L G

5 L L L L G6 L G G L7 G L G L8 L L L G L9 G G L L

10 L L G L L11 L G L L L12 G L L L L13 L L L L L L


10/38

SAKHY

( .)The number of metrical forms Snin the n-mtrprastra is the sum of thenumber of metrical forms Sn-1, Sn-2, in the prastras of n-1 and n-2 mtrsrespectively:

Sn= Sn-1+ Sn-2

The above rule follows from the fact that the n-mtrprastra is generatedas follows: The first Sn-2rows are obtained by adding a G to the right ofeach row of the prastra of (n-2)-mtrs. The next Sn-1rows are obtainedby adding an L to the right of each row of the prastra of (n-1)-mtrs.

Since the prastra of one mtrhas only one metrical form and that of 2-mtrs has just two metrical forms (G and LL), we get the followingsakhyka sequence (the so called Fibonacci (c.1200) sequence)

n 1 2 3 4 5 6 7 8 9 10Sn 1 2 3 5 8 13 21 34 55 89


11/38


12/38

NAA

(.-)

To find the moric metric form associated with a given row-number in the

n-mtr-prastra

Write down nLs with the sequence of sakhykas above them.

Subtract the given row number from the sakhyka Sn.

From the result, subtract Sn-1if possible. Otherwise, subtract Sn-2and

so on till the end. The moric metric form is obtained by converting each L below a

sakhyka which has been subtracted, together with the L to theright of it, into a G.


13/38

NAA

Example: To find the seventh metrical form in the 6-MtrPrastra

1 2 3 5 8 13L L L L L L

13-7 = 6.

8 cannot be subtracted form 6

6-5 = 1 3, 2 cannot be subtracted from 1

1-1= 0

Thus 5 and 1 are the sakhykas which have been subtracted.

Hence the metric form is GLGL

The above examples are based upon the representations

9 = 1 + 8 and 6 = 1+5

In fact it can be shown the naa and uddia processes are based on thevery interesting property that every integer can be expressed uniquely as a

sum of non-consecutive Virahka (Fibonacci) numbers.


14/38

MTR-MERU

The number of metric forms which different number of Laghus (Gurus)

can be found from the following table known as the mtr-meru:

(.-)

The successive rows of the mtr-meru have 1, 2, 2, 3, 3, etc cells.

Place 1 in the top row and in the end of each row.

Place 1 in the beginning of all the even rows and 2, 3, 4 etc in thebeginning of odd rows.

The other cells are filled by the sum of the number in the row above whichis above or to the left and the number in the row further above which is to

the right.


15/38


16/38

PRATYAYAS IN SAGTA-RATNKARA

The study of combinatorial questions in music was undertaken by

rgadeva (c.1225) in his celebrated treatise on music, Sagta-

ratnkara. In the first chapter of Sagta-ratnkara there is a discussion

of Tna-Prastra which generates all the possible tnas that can be formedfrom the seven svaras. Later, in Chapter V, there is a very elaborate

discussion of the more complicated Tla-Prastra.

In tna-prastra, rgadeva considers permutations or tnas of subsets of

the seven basic musical notes which we denote as S, R, G, M, P, D, N.The sakhyor the total number of rows in the prastra is the factorial of

the number of elements in tna.

Example: Tna-prastra of S R G

1 S R G2 R S G

3 S G R4 G S R5 R G S6 G R S


17/38

TNA-PRASTRA

: ( ..-)

The first row has all the svaras in the original order. Successive lines inthe prastra are generated as follows.

Starting from the left, identify the first svara which has at least onelower svara to the left. Below that is placed the highest of these (lower)svaras.

Then the svaras to the right are brought down as they are.

The svaras left out are placed in the original order to the left, thus

completing the next line of the prastra.

rgadevas rule for the construction of the prastra is in general

applicable for the enumeration of the permutations of ndistinct elements

with a natural order. It generates all the permutations in the so-called

colex order (mirror image of lexicographic order in reverse).


18/38

EXAMPLE: TNA-PRASTRA OF SRGM

1 S R G M2 R S G M3 S G R M4 G S R M5 R G S M6 G R S M7 S R M G8 R S M G9 S M R G

10 M S R G11 R M S G12 M R S G13 S G M R14 G S M R15 S M G R16 M S G R17 G M S R18 M G S R19 R G M S20 G R M S

21 R M G S22 M R G S23 G M R S24 M G R S


19/38

KHAA-MERU

In order to discuss the naa and uddia processes, rgadeva introduces

the so called Khaa-Meru::

:

( ..-)

Place 1 followed by 0-s in the first row.

Place the factorials of 1, 2, 3 etc., in the next row, starting from the

second column.

Place twice, thrice etc., of the factorials in the succeeding rows, starting

from a later column at each stage.


20/38

KHAA-MERU

S R G M P D N1 0 0 0 0 0 0

1 2 6 24 120 7204 12 48 240 1440

18 72 360 2160

96 480 2880600 3600

4320

Note that starting from the second row, each column consists of themultiples of factorials. As we shall see they play a crucial role in the naa

and uddia processes.


21/38

UDDIA

( ..-)

Given a tna (of n svaras), note the rank of the last svara in the reverse

of natural order among the given svaras. Mark the corresponding entryin the last or the n-th column.

Note the rank of the next svara (in the reverse of natural order) among

the remaining svaras. Mark the corresponding entry in the next or the

(n-1)-th column. And so on. The uddia or the rank-number of the given tna will be the sum of the

all the marked entries.


22/38

UDDIA

Example: To find the row of the tna MSRG in the prastra of SRGM

G is the second from the last among SRGM. Hence, mark 6, the entryin second row, in the last or the fourth column.

R is the second from the last among the remaining SRM. Hence Mark

2 in the third column.

S is the second from the last among SM. Hence, mark 1 in the secondcolumn.

M is the only svara left. Mark 1 in the first column.

Row-number of MSRG in the prastra = 1+ 1+ 2+ 6 = 10

S R G M1 0 0 0

1 2 6

4 1218


23/38

NAA

( ..-)

To find the tna (of n svaras) corresponding to a given row-number(naa-sakhy), mark the entry just below the rank-number in the n-th

column.

Subtract that entry from the rank number and mark the entry, which is

just below the resulting number, in the next or the (n-1)-th column. Andso on.

The position of the marked entry in the last column gives the rank of the

last svara of the tna in the reverse natural order.

The position of the marked entry in the next column gives the rank ofthe last but one svara, amongst the remaining svaras, in the reverse

natural order.


24/38

NAA

Example: To find the 18thtna in the prastra of SRGM

In the fourth column, the number just below 18 is 2, which is in thethird row. Hence, the fourth svara is the third among SRGM inreverse order: R

18-12 = 6. In the third column, the number just below 6 is 4, which isin the third row, which is just below 6. Hence, the next svara is the

third among SGM in reverse order: S 6-4 = 2. In the second column, the number just below 2 is 1, which is

in the second row. Hence the next svara is the second among GM inreverse order: G

2-1 = 1. The other svara left is M

The 18thtna is MGSR

S R G M1 0 0 0

1 2 64 12

18


25/38

FACTORIAL REPRESENTATION OF RGADEVA

The naa and uddia processes are essentially based on a certain factorial

representation of numbersIn the above examples,

10 = 1+ 1+ 2 + 6 = 1x0! + 1x1! + 1x2! + 1x3!

18 = 1+ 1+ 4+ 12 = 1x0! + 1x1! + 2x2! + 2x3!

In fact the general result may be stated as follows:

Every integer 1m n! can be uniquely represented in the form

m = d00! + d11! + d22! + ... + dn-1(n-1)!,

where diare integers such thatd0= 1 and 0 di i, for i = 1, 2, ..., n-1.


26/38

TLA-PRASTRA

Chapter V of Sagtaratnkara is the Tldhyya with 409 verses. The

first 311 verses discuss mrga-tlas and about 120 de-tlas. At the end ofthis discussion, it is noted that there are indeed very many such tlas and it

would not be possible to display all of them. This sets the stage for the

prastra-prakaraa which takes up the remaining nearly 100 verses of theTldhyya.

The tlgas considered here are Druta, Laghu, Guru and Pluta, which aretaken to be of duration 1, 2, 4 and 6 respectively, in Druta units. Tla-

prastra consists in a systematic enumeration of all tlas with the same

total duration (kla-prama)

Thus the Tla-prastra is a non-trivial generalisation of Mtr-vtta-

prastra. Nryaa Paita in his Gaitakaumud(C.1350) has discussed

the simpler generalisation of Mtr-vtta-prastra, which involves only theelements L, G and P with relative values 1, 2 and 3 respectively.


27/38

TLA-PRASTRA

( .-)

rgadevas procedure for the construction of prastrais as follows:

The last row of the prastra has all drutas only.

In the first row, place as many Ps as possible to the right, followed, ifpossible (from right to left), by a G and a D or a G alone, or by an L and

a D or an L alone, or by a D alone, to the left.

To go from any row of the prastra to the next, identify the first non-Delement from the left. Place below that the element next to it induration: D below a L, L below a G and G below a P.

Bring down the elements to the right as they are.

Make up for the deficient units (if any) by adding to the left as many Psas possible, followed similarly by Gs, Ls and Ds in that order from rightto left.


28/38

6-DRUTA-PRSTRA

1 P 6

2 L G 2 43 D D G 1 1 44 G L 4 25 L L L 2 2 26 D D L L 1 1 2 2

7 D L D L 1 2 1 28 L D D L 2 1 1 29 D D D D L 1 1 1 1 2

10 D G D 1 4 111 D L D D 1 2 2 1

12 L D L D 2 1 2 113 D D D L D 1 1 1 2 114 G D D 4 1 115 L L D D 2 2 1 116 D D L D D 1 1 2 1 1

17 D L D D D 1 2 1 1 118 L D D D D 2 1 1 1 119 D D D D D D 1 1 1 1 1 1


29/38

7-DRUTA-PRSTRA

1 D P

2 D L G3 L D G4 D D D G5 D G L6 D L L L

7 L D L L8 D D D L L9 G D L

10 L L D L11 D D L D L

12 D L D D L13 L D D D L14 D D D D D L


30/38

7-DRUTA-PRSTRA (CONTD)

15 P D

16 L G D17 D D G D18 G L D19 L L L D20 D D L L D

21 D L D L D22 L D D L D23 D D D D L D24 D G D D25 D L L D D

26 L D L D D27 D D D L D D28 G D D D29 L L D D D30 D D L D D D

31 D L D D D D32 L D D D D D33 D D D D D D D


31/38

SAKHY

( ., , )

rgadeva makes the observation that among all the tla-forms which

appear in the n-druta-prastra, Sn-1end in a D, Sn-2in a L, Sn-4in a G and

Sn-6 end in a P, and the total number of forms tla-forms Snin the n-druta-prastra is just the sum of these four numbers. Thus,

Sn= Sn-1+ Sn-2+ Sn-4+ Sn-6

Noting S1= 1, S2= 2, we get the rgadeva sequence of sakhykas:

n 1 2 3 4 5 6 7 8 9 10Sn 1 2 3 6 10 19 33 60 106 169

UDDIA


32/38

UDDIA

To find the row-number of a given tla-form in a n-Druta-Prastra, write

the rgadeva sakhykas S1, S2, ... sequentially from the left on top ofthe tla-form in the following way:

Write one sakhyka above a D, 2 above an L, 4 above a G and 6above each P.

Sum the following (we shall see later that these are what are calledthe patita-sakhykas): All the first sakhykas above the L, the

second and third sakhykas above the Gs and the second, fourth

and fifth sakhykas above the Ps.

The row-number of the given tla form is obtained by subtracting the

above sum from Sn.

UDDIA


33/38

UDDIA

Example: To find the row-number of LDLL in 7-druta-prastra

Sn 1 2 3 6 10 19 33L D L L

Total of the patita Sn: 19+6+1 = 26. Row-number: 33-26=7

Example: To find the row-number of GDLin 7-druta-prastra

Sn 1 2 3 6 10 19 33G D L

Total of the patita Sn: 19+3+2 = 24. Row-number: 33-24=9

Example: To find the row-number of PDin 7-druta-prastra

Sn 1 2 3 6 10 19 33

P D

Total of the patita Sn: 10 + 6+2 = 18. Row-number: 33-18 = 15

NAA


34/38

NAA

If it is required to find the tla-form in the r-th row of n-druta-prastra, the

following procedure is prescribed:

Place sequentially the rgadeva sequence of sakhykas S1, S2, ... Sn. Check if the sakhyka Sn-1 can be subtracted from (Sn-r). If so, mark

Sn-1 as p(patita-sakhyka) and go on to check if Sn-2 can be subtracted

from (Sn-r-Sn-1) and so on. If Sn-1 cannot be subtracted from (Sn-r) mark it as a (apatita-

sakhyka) and go on to check if Sn-2 can be subtracted from (Sn-r)

and so on.

In this way mark all sakhykas as either p ora.

Use the following signatures of various tlgas to find the tla-form:

Sn-6 Sn-5 Sn-4 Sn-3 Sn-2 Sn-1 SnD (a) aL (a) p aG (a) a p p aP (a) a p a p p a

NAA


35/38

NAA

Example: To find the 8thtla-form in the 7-druta-prastra:

33-8 = 25, 25-19 = 6, 6-6 = 0The patita and apatita Snare given below

p/a a a a p a p aSn 1 2 3 6 10 19 33

Starting from 33, since 19 is patita and 10 is apatita, we get an L atthe right extreme.

Starting from 10, since 6 is patita and 3 is apatita, we get another Lto the left of the first.

Starting from 3, since 2 is apatita, we get a D to the left.

Starting from 2, since 1 is apatita we get one more D to the left.

Since 1 is apatita, we get one more D.Thus the tla-form is DDDLL

NAA


36/38

NAA

Example: To find the 28thtla-form in the 7-druta-prastra:

33-28 = 5, 5-3 = 2, 2-2 = 0

p/a a p p a a a aSn 1 2 3 6 10 19 33

33, 19, 10 and 6 are apatita and thus we get DDD from the right.

Starting from 6, 3 and 2 are patita and 1 is apatita. They give a G.

Thus the tla-form is GDDD

It can be shown that both the naa and uddia processes for the tla-

prastra are based on a very interesting property that every natural integer

can be uniquely written as a sum of the rgadeva sakhykas S1, S2, ...,

satisfying certain conditions.

LAGHU-MERU


37/38

LAGHU-MERU

rgadeva discusses the lagakriy process for Tla-prastra in term ofvarious tables, Druta-meru, Laghu-Meru, Guru-Meru and Pluta-Meru. We

display below the Laghu-Meru:

11 5 15

1 4 10 20 391 3 6 10 18 33 61

1 2 3 4 7 12 21 34 541 1 1 2 3 5 7 10 14 21

1 2 3 4 5 6 7 8 9 10

For instance, the column 7 of the above Meru shows that of the 19 tla-forms in the 7-Druta-prastara there are 7 tla-forms with 0L, 12 with 1L,

10 with 2L and 4 with 3L.

PRASTRA AND REPRESENTATION OF NUMBERS


38/38

PRASTRA AND REPRESENTATION OF NUMBERS

These instances of prastras in prosody and music show that in each case

there is associated a unique representation of the natural integers in terms

of the sakhykas associated with the prastra. It is this representationwhich facilitates the naa and uddia processes in each of these prastras.

The vara-vtta prastrahas associated with it the binary representation

of natural numbers.

The mtr-vtta-prastra has associated with it a representation ofnumbers in terms of Virahka-Fibonacci numbers.

The tna-prastra of rgadeva has associated with it the factorial

representation of numbers.

The tla-prastaraof rgadeva has associated with it a representation of

numbers in terms of rgadeva numbers.

The prastra of combinationsof robjects selected from a set of n, has

been studied by Nryaa Paita in Gaitakaumd. Here, the

sakhykas are the binomial coefficients nCr and there is an associatedrepresentation of every number as a sum of such binomial co-efficients.

03 combinatorics

Documents