Permutations and Oblong Numbers in the Theravāda-vinaya: A New Intersection of Buddhism and Indian Mathematics

Abstract

Within the context of Indian religions, Jainism has long been recognized for its extensive use of permutations and combinations. However, the application of these principles within Buddhist scriptures has received relatively little scholarly attention. This paper introduces a new example of the specific application of permutations and combinations in Buddhist scriptures. In this paper, we focus on the first saṅghādisesa rule in the Theravāda-vinaya, which lists a series of element sets and arranges these elements according to a certain pattern known as “ten-roots” (mūla), and we discover that these arrangements form a regular numerical sequence, called “oblong numbers”. Moreover, similar patterns with different quantities are also found in the fourth Pārājika and the fifth saṅghādisesa rules. This indicates that the compilers of the Theravāda-vinaya did not use this mathematical knowledge without basis. Interestingly, we also found the use of this sequence in the Bakhshālī manuscript. Therefore, in this article, after summarizing and verifying the arrangement rules of the Theravāda-vinaya, we discuss whether the oblong numbers were influenced by Greek mathematics.

1. Introduction

The origin of Indian mathematics has always been a controversial topic among scholars. Due to the dominance of Eurocentrism for centuries, some scholars believed that Greece was the source of modern science and that Europe inherited Greek culture.1 This led to the neglect of the mathematical development of other regions. This belief prompted them to think that Indian mathematics was largely influenced by Greek mathematics. For example, Kaye looked for traces of Greek mathematics in the Bakhshālī Manuscript and insisted that the Bakhshālī manuscript was written no earlier than the 12th century. He also listed some examples of Greek mathematical influence on Indian mathematics, but these have been refuted by scholars (Hayashi 1995, pp. 132–33).

As Heeffer (2010) pointed out, as early as the late 19th century, the German scholar M. Cantor (1880–1908) expressed that “the Indians learned algebra through traces of algebra within Greek geometry” and “Brahmagupta’s solution to quadratic equations has Greek origins.”2 However, these views were strongly refuted by Herman Hankel (1839–1873), as follows:

That by humanist education deeply inculcated prejudice that all higher intellectual culture in the Orient, in particular all science, is risen from Greek soil and that the only mentally truly productive people have been the Greek, makes it difficult for us to turn around the direction of influence for one instant.3

On the other hand, many scholars hold a “non-Eurocentric” stance; that is to say, they have come to acknowledge the independence and originality of Indian mathematics. They suggest that the origins of Indian mathematics can be traced back to the Harappan civilization of the third millennium BCE, regarding which archeologists have discovered that “the archaeological finds described below do provide some indication, however meager, of the nature of the numerate culture that this civilization possessed” (Joseph 2010, pp. 317–21). These include a number of different plumb bobs of uniform size and weight and scales and instruments for measuring length (Merzbach and Boyer [1968] 2011, p. 186; Katz 2009, p. 231). Throughout the subsequent extended periods, Indian mathematics saw significant developments in religion, music, medicine, architecture, and astronomy, among other areas.

Regarding Buddhism and Indian mathematics, many scholars have discussed the use of large numbers and units in Buddhist scriptures (Niu 2004, pp. 22–32) and the concepts of “śūnya” (emptiness) and zero (Joseph 2010, p. 345), which are also featured in the scriptures of Hinduism and Jainism (Joseph 2010, pp. 338–47). Additionally, many scholars have focused more on Buddhism and astronomy, such as (Kotyk 2017, 2020; Niu 2004; Goble 2019; Yano 2013). Thus far, very few scholars have discussed specific cases in which mathematical knowledge is utilized in Buddhist scriptures. This article aims to provide some new cases showing the use of permutations and combinations from mathematics in the Vinaya text.

The mathematical case discussed in this paper is drawn from the Vinaya texts in Buddhism. The Vinaya “refers to the body of teachings concerning monastic discipline or law attributed to the historical Buddha” (Clarke 2015, p. 60), and the Theravāda-vinaya is the pāli Canon of vinayas, which, according to the Sri Lankan chronicle Mahāvaṃsa, was committed to writing and commentaries on it were produced in the first century BCE, thus probably predating the redactional closure of the other Vinayas (Kieffer-Pülz 2014, pp. 50–52).

This paper focuses on the first saṅghādisesa rule in the Theravāda-vinaya, which lists a series of element sets and arranges these elements according to a certain pattern known as “ten-roots” (mūla). After reconstructing all the arrangements, in this paper, we discover that these arrangements form a regular numerical sequence. Moreover, similar patterns with arrangements and combinations are also found in the fourth pārājika, the fifth saṅghādisesa rules of the Theravāda-vinaya, as well as in the fourth pārājika of Shisong lü. Therefore, in this article, we will summarize and verify the arrangement and combination rules in these vinaya texts, and we will discuss the possible origins of these arrangements and combinations.

2. The Permutation of the First Saṅghādisesa Rule in the Theravāda-vinaya

The first Saṅghādisesa rule in the Theravāda-vinaya prohibits a monk from intentionally ejaculating, which is specified as follows:

Intentional emission of semen, except during a dream, is an offense of Saṅghādisesa4.

In the explanation of this rule, the Theravāda-vinaya lists ten methods and reasons for ejaculation, ten purposes of ejaculation, and ten colors of semen. Regarding the ten purposes of ejaculation, the Theravāda-vinaya presents a unique arrangement from one root to ten roots. The ten purposes of ejaculation are as follows:

①

ārogyatthāya (for health)

②

sukhatthāya (for experience pleasure)

③

bhesajjatthāya (for medicinal purposes)

④

dānatthāya (for giving)

⑤

puññatthāya (for beneficial practices)

⑥

yaññatthāya (for festivals)

⑦

saggatthāya (to be born in a heaven)

⑧

bījatthāya (for seed)

⑨

vīmaṃsatthāya (to try)

⑩

davatthāya (for fun)5

Following this, the Theravāda-vinaya lists the arrangement methods from one-root to ten-root cases, but they are presented in an abbreviated form. In this article, efforts will be made to reconstruct the complete permutations from one-root to ten-root cases.

2.1. One-Root Permutation (Ekamūlaka)

First, let us examine the one-root permutation called ekamūlakassa khaṇḍacakka6. It involves combinations of two elements. Specifically, combinations of element ① with other elements leading to a total of nine forms are listed below:

① Ārogyatthañca ② sukhatthañca

① Ārogyatthañca ③ bhesajjatthañca

① ārogyatthañca ④ dānatthañca

① ārogyatthañca ⑤ puññatthañca

① ārogyatthañca ⑥ yaññatthañca

① ārogyatthañca ⑦ saggatthañca

① ārogyatthañca ⑧ bījatthañca

① ārogyatthañca ⑨ vīmaṃsatthañca

① ārogyatthañca ⑩ davatthañca

Next, we move to the ekamūlakassa baddhacakka7 segment. The pairs start with element ② sukhatthāya combining with ③ bhesajjatthāya, and end with the combination of ② sukhatthāya and ① ārogyatthāya. The nine types are as follows:

② Sukhatthañca ③ bhesajjatthañca

② Sukhatthañca ④ dānatthañca

② sukhatthañca ⑤ puññatthañca

② sukhatthañca ⑥ yaññatthañca

② sukhatthañca ⑦ saggatthañca

② sukhatthañca ⑧ bījatthañca

② sukhatthañca ⑨ vīmaṃsatthañca

② sukhatthañca ⑩ davatthañca

② Sukhatthañca ① ārogyatthañca

Thereafter, combinations of element ③ bhesajjatthāya start with merging with ④ dānatthāya and end with merging with ② sukhatthāya, yielding nine varieties:

③ Bhesajjatthañca ④ dānatthañca……

③ bhesajjatthañca ⑤ puññatthañca…

③ bhesajjatthañca ⑥ yaññatthañca…

③ bhesajjatthañca ⑦ saggatthañca…

③ bhesajjatthañca ⑧ bījatthañca…

③ bhesajjatthañca ⑨ vīmaṃsatthañca…

③ bhesajjatthañca ⑩ davatthañca

③ Bhesajjatthañca ① ārogyatthañca…

③ bhesajjatthañca ② sukhatthañca

Following this, similar types of combinations are formed between each pair, leading to a total of ninety forms, which can also be viewed as a permutation of a set (ten elements) two at a time. Using the permutation formula, P(10, 2) = 90.

{④, ⑤}, {④, ⑥}, {④, ⑦}, {④, ⑧}, {④, ⑨}, {④, ⑩}, {④, ①}, {④, ②}, {④, ③}……

{⑩, ①}, {⑩, ②}, {⑩, ③}, {⑩, ④}, {⑩, ⑤}, {⑩, ⑥}, {⑩, ⑦}, {⑩, ⑧}, {⑩, ⑨}

2.2. Two-Root Permutation (Dumūlaka)

Two-root permutation in the Theravāda-vinaya:

① Ārogyatthañca ② sukhatthañca ③ bhesajjatthañca ceteti upakkamati muccati, āpatti saṅghādisesassa…pe… ① ārogyatthañca ② sukhatthañca ⑩ davatthañca ceteti upakkamati muccati, āpatti saṅghādisesassa.

Dumūlakassa khaṇḍacakkaṃ.

② Sukhatthañca ③ bhesajjatthañca ④ dānatthañca ceteti upakkamati muccati, āpatti saṅghādisesassa…pe… ② sukhatthañca ③ bhesajjatthañca ⑩ davatthañca…pe… ② sukhatthañca ③ bhesajjatthañca ① ārogyatthañca ceteti upakkamati muccati, āpatti saṅghādisesassa.

Dumūlakassa baddhacakkaṃ saṃkhittaṃ

⑨ Vīmaṃsatthañca ⑩ davatthañca ① ārogyatthañca ceteti upakkamati muccati, āpatti saṅghādisesassa. ⑨ Vīmaṃsatthañca ⑩ davatthañca ⑧ bījatthañca ceteti upakkamati muccati, āpatti saṅghādisesassa.

Dumūlakaṃ niṭṭhitaṃ.8

In the two-root permutation, the first step is to pair the two elements to form a unit; for example, Elements ① and ② form the unit ①②. This unit is sequentially combined with each of the elements from ③ to ⑩, resulting in a total of eight different forms. This rule is called khaṇḍacakka.

Next, Element ② is paired with ③ to form the unit ②③. This pair is then sequentially combined with every element from ④ to ⑩, as well as ①, also resulting in a total of eight different forms.

Finally, Element ⑨ is paired with ⑩ to form the unit ⑨⑩. Then, this pair is sequentially combined with each of the elements from ① to ⑧, also resulting in eight forms. This rule is called baddhacakka. It is noteworthy that there is no type where ⑩ is paired with ① and then combined with other elements. It can be considered that each element appears only once.

In the two-root case, there are nine types, each type has eight different forms, totaling seventy-two different forms. This is different from the number of combinations and permutations achieved by choosing three elements from 10, which would be C(10, 3) = 120 and P(10, 3) = 720.

However, we can still discern certain patterns. First, the arrangement can be divided into two parts: the “fore element” can be a single element or a unit of elements, such as ①② in the two-root case, while the “after element” consists of only one element.

Second, the sequence of choosing fore elements adheres to an order from ① to ⑩. For example, ① and ② or ② and ③ can form a unit, but ⑩ and ① cannot form a unit.

Third, numbers do not repeat within a combination. For instance, in the two-root scenario, the unit made up of ⑦ and ⑧ acts as the fore element, initially combining sequentially with ⑨ or ⑩, then with ①, until it finally combines with ⑥, without recombining with elements ⑦ or ⑧. This is referred to as baddhacakka. For the two roots group, a configuration like {⑨, ⑩; ⑧} is possible, while {⑨, ⑩; ⑨} is not.

Based on these characteristics, we can consider the rules of “two roots” as a permutation calculation within a set of these ten elements, where initially, two consecutive elements are chosen as fore elements, and then they are combined with other elements. This process essentially equates to a permutation calculation selecting two elements from a set of nine elements. For example, initially selecting elements ① and ② as fore elements and then combining them with the after element ranging from ③ to ⑩ results in P (9,2) = 72, which matches the previously computed number. Therefore, we can infer that the method of root calculation in the Theravāda-vinaya can be regarded as P(10−r + 1, 2), with r representing the number of roots.

2.3. The Conjecture from Three Roots (Timūlaka) to Ten Roots (Sabbamūlaka)

The Theravāda-vinaya lacks a comprehensive explanation of the transition from three-root to ten-root configurations.9 Nevertheless, we can project these permutations using characteristics and formulas derived from one-root and two-root transformations. Additionally, these calculations will be presented through a visualization of the permutation process.

For permutations involving three-root, the process begins with the selection of three sequential elements defined as fore elements, which are then paired with other elements:

{①, ②, ③; ④}, {①, ②, ③; ⑤},…,{①, ②, ③; ⑩} (seven forms in total)

{②, ③, ④; ⑤},…,{②, ③, ④; ⑩}, {②, ③, ④; ①}

{③, ④, ⑤; ⑥},…,{③, ④, ⑤; ⑩}, {③, ④, ⑤; ①}, {③, ④, ⑤; ②}

{④, ⑤, ⑥; ⑦},…,{④, ⑤, ⑥; ⑩}, {④, ⑤, ⑥; ①},…,{④, ⑤, ⑥; ③}

{⑤, ⑥, ⑦; ⑧},…,{⑤, ⑥, ⑦; ⑩}, {⑤, ⑥, ⑦; ①},…,{⑤, ⑥, ⑦; ④}

{⑥, ⑦, ⑧; ⑨}, {⑥, ⑦, ⑧; ⑩}, {⑥, ⑦, ⑧; ①},…,{⑥, ⑦, ⑧; ⑤}

{⑦, ⑧, ⑨; ⑩}, {⑦, ⑧, ⑨; ①},…,{⑦, ⑧, ⑨; ⑥}

{⑧, ⑨, ⑩; ①}, {⑧, ⑨, ⑩; ②},…,{⑧, ⑨, ⑩; ⑦}

Calculations for the three roots system yield fifty-six unique permutations, derived via the permutation formula calculation: P(8, 2) = 56, confirming the consistency of both forms.

For four roots, the permutations result in forty-two distinct arrangements, calculated as P(7, 2) = 42, demonstrated by configurations such as:

{①, ②, ③, ④; ⑤}, {①, ②, ③, ④; ⑥},…,{①, ②, ③, ④; ⑩} (six forms)

{②, ③, ④, ⑤; ⑥},…,{②, ③, ④, ⑤; ⑩}, {②, ③, ④, ⑤; ①}

{③, ④, ⑤, ⑥; ⑦},…,{③, ④, ⑤, ⑥; ⑩}, {③, ④, ⑤, ⑥; ①}, {③, ④, ⑤, ⑥; ②}

{④, ⑤, ⑥, ⑦; ⑧},…,{④, ⑤, ⑥, ⑦; ⑩}, {④, ⑤, ⑥, ⑦; ①},…,{④, ⑤, ⑥, ⑦; ③}

{⑤, ⑥, ⑦, ⑧; ⑨}, {⑤, ⑥, ⑦, ⑧; ⑩}, {⑤, ⑥, ⑦, ⑧; ①},…,{⑤, ⑥, ⑦, ⑧; ④}

{⑥, ⑦, ⑧, ⑨; ⑩}, {⑥, ⑦, ⑧, ⑨; ①},…,{⑥, ⑦, ⑧, ⑨; ⑤}

{⑦, ⑧, ⑨, ⑩; ①},…,{⑦, ⑧, ⑨, ⑩; ⑥}

Calculating for five-root gives thirty permutations according to P(6, 2) = 30.

{①, ②, ③, ④, ⑤; ⑥},…,{①, ②, ③, ④, ⑤; ⑩} (five forms)

{②, ③, ④, ⑤, ⑥; ⑦},…,{②, ③, ④, ⑤, ⑥; ⑩}, {②, ③, ④, ⑤, ⑥; ①}

{③, ④, ⑤, ⑥, ⑦; ⑧},…,{③, ④, ⑤, ⑥, ⑦; ⑩}, {③, ④, ⑤, ⑥, ⑦; ①}, {③, ④, ⑤, ⑥, ⑦; ②}

{④, ⑤, ⑥, ⑦, ⑧; ⑨},…,{④, ⑤, ⑥, ⑦, ⑧; ⑩}, {④, ⑤, ⑥, ⑦, ⑧; ①},…,{④, ⑤, ⑥, ⑦, ⑧; ③}

{⑤, ⑥, ⑦, ⑧, ⑨; ⑩}, {⑤, ⑥, ⑦, ⑧, ⑨; ①},…,{⑤, ⑥, ⑦, ⑧, ⑨; ④}

{⑥, ⑦, ⑧, ⑨, ⑩; ①},…,{⑥, ⑦, ⑧, ⑨, ⑩; ⑤}

Six roots result in twenty variations, calculated as P(5, 2) = 20:

{①, ②, ③, ④, ⑤, ⑥; ⑦},…,{①, ②, ③, ④, ⑤, ⑥; ⑩} (four forms)

{②, ③, ④, ⑤, ⑥, ⑦; ⑧},…,{②, ③, ④, ⑤, ⑥, ⑦; ⑩}, {②, ③, ④, ⑤, ⑥, ⑦; ①}

{③, ④, ⑤, ⑥, ⑦, ⑧; ⑨},…,{③, ④, ⑤, ⑥, ⑦, ⑧; ①}, {③, ④, ⑤, ⑥, ⑦, ⑧; ②}

{④, ⑤, ⑥, ⑦, ⑧, ⑨; ⑩},…,{④, ⑤, ⑥, ⑦, ⑧, ⑨; ①},…,{④, ⑤, ⑥, ⑦, ⑧, ⑨; ③}

{⑤, ⑥, ⑦, ⑧, ⑨, ⑩; ①},…,{⑤, ⑥, ⑦, ⑧, ⑨, ⑩; ④}

For seven roots, there are twelve variations, found by P(4, 2) = 12:

{①, ②, ③, ④, ⑤, ⑥, ⑦; ⑧}, {①, ②, ③, ④, ⑤, ⑥, ⑦; ⑨}, {①, ②, ③, ④, ⑤, ⑥, ⑦; ⑩} (three forms)

{②, ③, ④, ⑤, ⑥, ⑦, ⑧; ⑨}, {②, ③, ④, ⑤, ⑥, ⑦, ⑧; ⑩}, {②, ③, ④, ⑤, ⑥, ⑦, ⑧; ①}

{③, ④, ⑤, ⑥, ⑦, ⑧, ⑨; ⑩}, {③, ④, ⑤, ⑥, ⑦, ⑧, ⑨; ①}, {③, ④, ⑤, ⑥, ⑦, ⑧, ⑨; ②}

{④, ⑤, ⑥, ⑦, ⑧, ⑨, ⑩; ①}, {④, ⑤, ⑥, ⑦, ⑧, ⑨, ⑩; ②}, {④, ⑤, ⑥, ⑦, ⑧, ⑨, ⑩; ③}

Eight roots show six variations, calculated as P(3, 2) = 6:

{①, ②, ③, ④, ⑤, ⑥, ⑦, ⑧; ⑨}, {①, ②, ③, ④, ⑤, ⑥, ⑦, ⑧; ⑩} (two forms)

{②, ③, ④, ⑤, ⑥, ⑦, ⑧, ⑨; ⑩}, {②, ③, ④, ⑤, ⑥, ⑦, ⑧, ⑨; ①}

{③, ④, ⑤, ⑥, ⑦, ⑧, ⑨, ⑩; ①}, {③, ④, ⑤, ⑥, ⑦, ⑧, ⑨, ⑩; ②}

Nine roots come to two variations according to P(2,2) =2:

{①, ②, ③, ④, ⑤, ⑥, ⑦, ⑧, ⑨; ⑩} (one forms)

{②, ③, ④, ⑤, ⑥, ⑦, ⑧, ⑨, ⑩; ①}

Finally, there is one configuration for the full complement of roots (an exception):

{①, ②, ③, ④, ⑤, ⑥, ⑦, ⑧, ⑨, ⑩}

In our analysis of the Theravāda-vinaya, we have identified distinct patterns in the arrangement of elements:

• Within the Theravāda-vinaya, the configurations are categorized as khaṇḍacakka and baddhacakka. Khaṇḍacakka refers to a sequence arranged naturally from ① to ⑩. Baddhacakka, on the other hand, includes arrangements such as those starting from ②, continuing to ⑩, then looping back to incorporate ①, or starting from ③, moving through to the set with ⑩, and then looping back to include ① and ②. This reflects the cyclic pattern inherent in baddhacakka.
• From two-root to nine-root configurations in the Theravāda-vinaya, each configuration can be viewed as a permutation of two elements drawn from distinct sets. For instance, the two-root configuration is perceived as a permutation of two elements selected from a pool of nine elements, and this pattern holds similarly for other configurations. The total number of permutations from oneroot to nine roots is calculated using the formula P(10-r + 1, 2), where r represents the number of roots. This is depicted in the following

  Table 1. The sum of the khaṇḍacakka and baddhacakka of the r-roots and the corresponding formulas in the first Saṅghādisesa rule.

  r-Root	SUM	Formula P(10 − r + 1, 2)
  one-root	90	P(10 − 1 + 1, 2)
  two-root	72	P(9, 2)
  three-root	56	P(8, 2)
  four-root	42	P(7, 2)
  five-root	30	P(6, 2)
  six-root	20	P(5, 2)
  seven-root	12	P(4, 2)
  eight-root	6	P(3, 2)
  nine-root	2	P(2, 2)
  ten-root	1	Exception

  .
• Interestingly, the counts of permutations conform to a pattern wherein each sum is the product of two consecutive non-negative integers, i.e., n × (n + 1), such as 2 = 12, 6 = 23, 12 = 34, 20 = 45. This series is known as oblong numbers.

Now, we will examine several of the annotations provided in Samantapāsādikā.

2.4. Commentary on the Theravāda-vinaya: Samantapāsādikā

The Samantapāsādikā is Buddhaghosa’s commentary on the Theravāda-vinaya from the 5th century.

First, concerning the concepts of khaṇḍacakka and baddhacakka, the commentary in the Samantapāsādikā states:

Tattha ārogyatthañca sukhatthañca ārogyatthañca bhesajjatthañcā ti evaṃ ārogyapadaṃ sabbapadehi yojetvā vuttamekaṃ khaṇḍacakkaṃ. Sukhapadādīni sabbapadehi yojetvā yāva attano attano atītānantarapadaṃ tāva ānetvā vuttāni nava baddhacakkānīti evaṃ ekamūlakāni dasa cakkāni honti. (Sp. III, p. 525).

Translation:

“ārogyatthañca sukhatthañca ārogyatthañca bhesajjatthañcā” (in the Theravāda-vinaya) states that combining the element ① ārogyattha (for health) with all other elements refers to khaṇḍacakka of one-root. Elements such as ② sukhattha (for pleasure) etc. when combined with all other elements up to the element preceding itself, characterize the nine forms of baddhacakkaṃ. These are referred to as the ten kinds of cakka based on one-root.10

Next, regarding the forms from one-root to ten-root configurations, the Samantapāsādikā explains the following:

idāni ārogyatthāyā’ti ādīsu tāva dasasu padesu paṭipāṭiyā vā uppaṭipāṭiyā vā heṭṭhā vā gahetvā upari gaṇhantassa, upari vā gahetvā heṭṭhā gaṇhantassa, ubhato gahetvā majjhe ṭhapentassa majjhe vā gahetvā ubhato gaṇharantassa sabbamūlakamṃ katvā gaṇhantassa cetanūpakkamamocane sati visaṅketo nāma natthīti dassetuṃ ārogyatthañ ca sukhatthañcā ‘ti khaṇḍacakkabaddhacakkādibhedaṃ vicittaṃ pāḷim āha. (Sp. III, p. 525).

Translation:

Regarding the ① ārogyattha (for health) etc. in the Theravāda-vinaya, and the previously mentioned ten phrases, it is possible to organize these phrases (1) in order, (2) in reverse order, or (3) starting from the bottom and moving progressively upward, (4) starting from the top and moving progressively downward, (5) starting from both ends and placing in the middle, (6) starting from the middle and extending to both sides, (7) or considering all roots in their entirety. If one engages in “thinking, acting, and then releasing,” there is no “exemption from offense” (visanketa). To illustrate this, the text categorizes various collections of khandacakka and baddhacakka, such as referring to ① ārogyattha (for health) and ② sukhattha (for pleasure).

This explanation also refers to a complex assortment of combinations ranging from having one root to ten roots, where the processing of sequences (1) and reverse sequences (2) or of full roots (7) are comprehensible. The concept of starting from below (3) or above (4) reflects similar thinking, while the condition of approaching from the edges to the middle (5) or conversely (6) might denote different integration tactics within the structural alignment of the root series. However, based on our presumed patterns, such an occurrence does not exist.

In summary, although the Samantapāsādikā describes some observable regularities, the lack of detailed explanations makes it challenging to fully comprehend its meaning. It is clear, however, that the Samantapāsādikā does not lay out precise computational results in its explanations.

3. The Permutation of the Fourth Pārājika Offense in the Theravāda-vinaya

The fourth pārājika offense in the Theravāda-vinaya concerns false claims of having attained the highest spiritual state. It includes thirty-one specific instances of the highest spiritual practices and exhibits a more complex arrangement compared to the first Saṅghādisesa offense.11

The Theravāda-vinaya lists the 31 elements as follows:

①

paṭhamaṃ jhānaṃ (First jhāna)

②

dutiyaṃ jhānaṃ (Second jhāna)

③

tatiyaṃ jhānaṃ (Third jhāna)

④

catutthaṃ jhānaṃ (Fourth jhāna)

⑤

suññato vimokkho (Void liberation)

⑥

animitto vimokkho (Signless liberation)

⑦

appaṇihito vimokkho (Desireless liberation)

⑧

suññato samādhi (Concentration on void)

⑨

animitto samādhi (Signless concentration)

⑩

appaṇihito samādhi (Desireless concentration)

⑪

suññatā samāpatti (Attainment of void)

⑫

animittā samāpatti (Signless attainment)

⑬

appaṇihitā samāpatti (Desireless attainment)

⑭

tisso vijjā (Three knowledges)

⑮

cattāro satipaṭṭhānā (Four foundations of mindfulness)

⑯

cattāro sammappadhānā (Four right efforts)

⑰

cattāro iddhipādā (Four bases of psychic power)

⑱

pañcindriyāni (Five faculties)

⑲

pañca balāni (Five powers)

⑳

satta bojjhaṅgā (Seven factors of enlightenment)

㉑

ariyo aṭṭhaṅgiko maggo (Noble Eightfold Path)

㉒

sotāpattiphalassa sacchikiriyā (Realization of stream-entry)

㉓

sakadāgāmiphalassa sacchikiriyā (Realization of once-returning)

㉔

anāgāmiphalassa sacchikiriyā (Realization of non-returning)

㉕

arahattassa sacchickiriyā (Realization of arahantship)

㉖

rāgassa pahānaṃ (Elimination of greed)

㉗

dosassa pahānaṃ (Elimination of aversion)

㉘

mohassa pahānaṃ (Elimination of delusion)

㉙

rāgā cittaṃ vinīvaraṇatā (Removal of greed from the mind)

㉚

dosā cittaṃ vinīvaraṇatā (Removal of aversion from the mind)

㉛

mohā cittaṃ vinīvaraṇatā (Removal of delusion from the mind)

3.1. The Khaṇḍacakka and Baddhacakka According to R-Root Cases

In the fourth Pārājika offense, the one-root khaṇḍacakka consists of individual combinations of ① and the remaining 30 elements, totaling 30 forms. Similarly, the one-root bad dhacakka involves combinations of ② with the remaining 30 elements, starting from ③ to 31, then looping back to incorporate ①, and so on, up to the 31st element combined with the other 30 elements, resulting in a total of 30 × 31 = 930 for one-root. Using the arrangement formula we previously speculated on, the result is calculated as P(31 − 1 + 1, 2) = 930, which matches our computation.

Subsequently, the Theravāda-vinaya omits the cases from two-root to thirty-one-root cases. However, based on the description for the one-root cases, we know that the structure of the one-root khaṇḍacakka and baddhacakka follows the same pattern as the first Saṅghādisesa offense. Consequently, we can extrapolate based on the pattern of the first Saṅghādisesa offense.

Table 2. The sum of the khaṇḍacakka and baddhacakka of the r-roots and the corresponding formulas in the fourth pārājika offense.

R-Root	SUM	Formula P(31 − r + 1, 2)
one-root	930	P(31, 2)
two-root	870	P(30, 2)
three-root	812	P(29, 2)
four-root	756	P(28, 2)
five-root	702	P(27, 2)
six-root	650	P(26, 2)
seven-root	600	P(25, 2)
eight-root	552	P(24, 2)
nine-root	506	P(23, 2)
ten-root	462	P(22, 2)
more than eleven roots	Not mentioned in the Theravāda-vinaya
all-root	1	Exception

presents the specific numbers of arrangements:

It is also noteworthy that numbers such as 930, 870, 812, and 756 manifest as oblong numbers. This indicates that the arrangement in both cases involves selecting two items from different sets, which ultimately reveals a pattern characteristic of oblong numbers. We now further explore the explanation provided by the Samantapāsādikā.

3.2. The Annotation of Samantapāsādikā

Regarding khaṇḍacakka and baddhacakka, the explanation given by the Samantapāsādikā aligns with that of the first Saṅghādisesa offense. Khaṇḍacakka consists of combinations of Element ① paṭhamaṃ jhānaṃ (first jhāna) with each of the subsequent 30 elements, while baddhacakka involves initially pairing Element ② dutiyaṃ jhānaṃ (second jhāna) in sequence with Elements ③ to 31, and then returning to the beginning to pair it with the Element ①. Within the one-root case, in addition to the khaṇḍacakka of Element ① paṭhamaṃ jhānaṃ, and the baddhacakka of Element ② dutiyaṃ jhānaṃ, there are twenty-nine other baddhacakka (aññānipi ekūnatiṃsa baddhacakkāni), equaling a total of thirty-one “types” (Sp. II, p. 497).

In terms of the calculations for two-root schema, three-root schema, and so on up to all-root schema, the Samantapāsādikā posits that the methodology for the one-root case applies uniformly, listing the respective combinations for each configuration: the two-root case has 29 “types”, the three-root case features 28 “types”, and so forth, decreasing by one for each additional root up to 30 roots, which present just one “type”, totaling 435 for the cases between two roots and thirty roots (Sp. II, p. 497).

However, it is necessary to clarify what exactly the 29 “types” or 28 “types” mentioned in the Samantapāsādikā refer to. Taking “two-root has 29 ‘types’” as an example, first, it can be ascertained that these “types” do not refer to the total number of arrangements. According to the rules of root, khaṇḍacakka and baddhacakka, the two-root case, means that ① and ② first form the fore elements, which are then combined one by one with the remaining 29 elements. This corresponds to the permutation calculation of selecting 2 elements from a set of 30, yielding 29 × 30 = 870 permutations. Therefore, the assertion by the Samantapāsādikā that there are only 29 “types” for two-root cases is clearly incorrect. Thus, these 29 “types” cannot represent the total number of arrangements. Second, in terms of quantity, these 29 “types” might refer to the number of arrangements for each group of khaṇḍacakka or baddhacakka in the two-root configuration (this possibility was suggested to me by Professor Sasaki Shizuka), or they might refer to the number of baddhacakka in the two-root configuration, as the Samantapāsādikā explains the number of khaṇḍacakka and baddhacakka for the one-root case immediately prior to this.

Therefore, it can be concluded that the explanations provided by the Samantapāsādikā do not represent the total number of arrangements for the r-root case. This appears to be a straightforward progressive decremental summation.

4. Incomplete Combinatory Methods in Shisong lü

In comparison to other Vinayas, it is only in the Shisong lü that we observe cases comparable to those in the Theravāda-vinaya, and this is especially evident within the fourth Saṅghādisesa offense. This offense prohibits a monk from seeking carnal desires from a woman by praising sexual intercourse, which is deemed the highest kind of ministration. In this context, the Shisong lü enumerates nine types of ministrations used by monks in flattering praises: 1上 (superior), 2大 (great), 3 勝 (excellent), 4巧 (skillful), 5 善(good), 6 妙 (wonderful), 7福 (blessed), 8好 (pleasing), and 9 快 (delightful), which are coupled with three distinct combinatorial methodologies.

The first type involves pairing “1上 (superior)” with each of the next eight qualifiers, followed by ‘2大 (great)’ with each of the seven remaining qualifiers, and so on. This pairing is similar to the one-root khaṇḍacakka in the Theravāda-vinaya, where it is sequentially paired with subsequent elements without looping back to the beginning to continue the pairing, as in the following:

{1, 2}, {1, 3}, {1, 4},…,{1, 9}, totaling eight forms;

{2, 3}, {2, 4}, {2, 5},…,{2, 9}, totaling seven forms;

{3, 4}, {3, 5}, {3, 6},…,{3, 9}, totaling six forms;

This pattern continues up to {8,9}, yielding thirty-six combinations.12

The second method aligns more closely with the khaṇḍacakka with a two-root technique from the Theravāda-vinaya, where “1, 2” serve as the fore element and are paired with each subsequent set of seven qualifiers:

{1, 2; 3}, {1, 2; 4}, {1, 2; 5}, {1, 2; 6},…,{1, 2; 9}, totaling seven forms;

{2, 3; 4}, {2, 3; 5}, {2, 3; 6}, {2, 3; 7},…,{2, 3, 9}, totaling six forms;

{3, 4; 5}, {3, 4; 6}, {3, 4; 7}, {3, 4; 8},…,{3, 4; 9}, totaling five forms;

Progressing thusly until {7, 8; 9}, aggregating to twenty-eight combinations.13

The third methodology integrates “1, 2, 3” as a collective base, with subsequent pairs stemming from sequential involvement from ‘4’ to ‘9’. Unlike the previous types, here, combinations start with ‘1’ and systematically incorporate more elements to construct the fore element, subsequently forming pairs for the khaṇḍacakka:

{1, 2, 3; 4}, {1, 2, 3; 5}, {1, 2, 3; 6},…, {1, 2, 3; 9}, totaling six forms

{1, 2, 3, 4; 5}, {1, 2, 3, 4; 6}, {1, 2, 3, 4; 7},…, {1, 2, 3, 4; 9}, totaling five forms

{1, 2, 3, 4, 5; 6}, {1, 2, 3, 4, 5; 7}, {1, 2, 3, 4, 5; 8}, {1, 2, 3, 4, 5; 9}, totaling four4 forms

{1, 2, 3, 4, 5, 6; 7}, {1, 2, 3, 4, 5, 6; 8}{1, 2, 3, 4, 5, 6; 9}, totaling three forms

{1, 2, 3, 4, 5, 6, 7; 8}{1, 2, 3, 4, 5, 6, 7; 9}, totaling two forms.14

Totaling to twenty distinctive forms.

Although the Shisong lü does not incorporate the r-root combination classification, it still shares similarities with the Theravāda-vinaya. For instance, in the Shisong lü, combination elements can also be divided into a fore element and an after element. The fore element may consist of a single element or a group of elements, such as the previously mentioned ‘1, 2, 3’, while the after element consists of only one element.

Additionally, the fore elements in the Shisong lü follow a sequential order from 1 to 9, akin to the r-root category found within the Theravāda-vinaya. This arrangement follows the rules of khaṇḍacakka strictly. Since the rules of baddhacakka are not implicated here, the combinatorial number for the Shisong lü can be calculated using the combinatorial formula C(n, 2). For instance, the first type consists of the selection of two elements from a set of nine, resulting in C(9, 2) = 36 combinations. The second type, where the fore elements are ‘1, 2’, involves combinations of two elements from a set of eight, yielding C(8, 2) = 28 combinations.

However, the third type of combination in the Shisong lü does not adhere to this logic. In the first and second types of combinations, the fore elements can be units starting from numbers other than 1, such as ‘2, 3’ or ‘3, 4’. However, in the third type of combination, the fore elements invariably start from 1. For instance, combinations are presented as {1, 2, 3, 4; 5} or {1, 2, 3, 4, 5; 6}, with no examples, like {2, 3, 4; 5} or {3, 4, 5, 6; 7}, that begin with a number other than 1. It is apparent that in the third type, the combinational rule is not complete. However, it is uncertain whether this inconsistency originates from the translation or from the original Sanskrit.

Additionally, in the Shisong lü, the number of combination types in the three categories, i.e., eight forms, seven forms, six forms, five forms, four forms, three forms, two forms, one form in the first type, decrease sequentially. The total sum is applicable to the sum formula of an arithmetic sequence, i.e., S = n(n + 1)/2. For instance, the first type of combination amounts to a sequence sum from 8 to 1, which is S = 36; the second type constitutes a sum from 7 to 1, which is S = 28; and the third type represents a sum from 6 to 2, which is S = 20. Interestingly, this aligns with the combination sum rules found in the interpretation of the fourth Pārājika of Samantapāsādikā.

Overall, the three types of combinations within the Shisong lü closely resemble the Khaṇḍacakka in the Theravāda-vinaya. Moreover, the first two types fully conform to the combinational calculation formula C(n, 2). Additionally, the Shisong lü also demonstrates features that are similar to those found in the Samantapāsādikā.

Additionally, in the Abhidharma-mahāvibhāṣā-śāstra (Apidamodapiposhalun), we can observe combination methods similar to those found in the Shisong lü.15

5. Permutations and Combinations in Ancient Indian Mathematics

The permutations presented in the Theravāda-vinaya are calculated according to two rules. First, there is the khaṇḍacakka, which combines elements sequentially from start to end, as well as the baddhacakka, which combines elements from the beginning to the end and then loops back to the start. Second, the configurations of roots must also be considered. However, it is important to note that regardless of the number of roots, the permutations can be seen as selecting two elements from an (n − r + 1)-set, with r representing the number of roots, leading to a formula such as P(n − r + 1,2), which is a standard formula in the Theravāda-vinaya.

Furthermore, the permutations for the r-root case in the Theravāda-vinaya demonstrate a systematic numeric sequence, the product sequence of two consecutive non-negative integers, and these numbers are called oblong numbers.

Lastly, unfortunately, while the Samantapāsādikā explains khaṇḍacakka and baddhacakka, it does not provide valuable reference information regarding the number of arrangements for elements within the sets.

The subsequent question that we must ask is as follows: what influences the permutations found in the Theravāda-vinaya? Additionally, are there other instances of the use of oblong numbers in Indian contexts?

In the Suśruta-saṃhitā, combinations of the six flavors—sweet, acidic, saline, pungent, bitter, and astringent—are enumerated (Chakravarti 1932, p. 81; Datta and Singh 1992, p. 232). It documents the calculation of various combinations, including combinations of one element up to combinations of all six elements. The calculations are specified as C(6, 1) = 6, C(6, 2) = 15, C(6, 3) = 20, C(6, 4) = 15, C(6, 5) = 6, and C(6, 6) = 1, leading to a total of 63 combinations. This method appears similar to the first and second types of combination described in the Shisong lü.

Prosody constitutes a significant component of the Vedas, and Pingala’s Chandaḥ Sūtra presents intriguing operations on syllable combinations, which can be elucidated using the commentary by Halāyudha (10th century). Initially, Pingala employs a long syllable (guru, abbreviated as g) and a short syllable (laghu, abbreviated as l). Subsequently, he establishes rules for composing different numbers of syllables. For instance, a monosyllable is represented as g or l, while two syllables can be combined to form gg, lg, gl, or ll, resulting in four combinations. Three syllables yield eight possible combinations: ggg, lgg, glg, llg, ggl, lgl, gll, and lll. Four syllables can produce sixteen combinations, and five syllables yield thirty-two combinations, and so forth (Dvivedi and Singh 2013, p. 247).

This combinatorial calculation can be understood as 2ⁿ, representing the sum of different combinations of g or l in n syllables. The results are expressed as C(n, 0) + C(n, 1) + C(n, 2) + … + C(n, n−1) + C(n, n) = 2ⁿ. For example, the number of combinations for four syllables is sixteen (2⁴), which includes one combination that is all g syllables (C(4, 4)), four combinations with one g syllable (C(4, 3)), six combinations with two g syllables (C(4, 2)), four combinations with three g syllables (C(4, 1)), and one combination with no g syllable (C(4, 0)). The total number of these different combinations is sixteen. When represented graphically as Figure 1, this process forms the Staircase of Mount Meru, also known as Pascal’s triangle or Yang Hui’s triangle (楊輝三角) (Ibid. pp. 261–65).

With respect to these combinations, three syllables (n = 3) are formed by combining the previous two-syllable (n − 1) combinations (2² = four types) with g and l (4 × 2 = 8). Similarly, four-syllable combinations (n = 3) are formed by combining the eight combinations of three syllables (n − 1) with g or l. Thus, these combinations can be conceptualized as consisting of a fore element and an after element, where the fore element is the combination of the previous n − 1 syllables, and the after element is a single syllable (g or l). This characteristic of preceding and succeeding elements is analogous to the r-root rules deduced in the Theravāda-vinaya.

Arrangements and combinations are also areas in which Jaina is specialized. The Bhagavati-sūtra, dating from between the 1st century BCE/1st century CE and the 3rd century CE16, is filled with numerous examples of permutations and combinations. Scholars have verified that these calculations align with modern formulas (Datta 1929, pp. 133–36). For example, two souls entering into the seven hells are stated as follows:

When lodged in one hell—7 forms

When distributed in two—42 forms

When in three—35 forms, total 84 forms (Lalwani 1985, pp. 37–38).

As scholars suggest, this issue can be resolved using modern formulas for permutations and combinations. For the calculation of “distributed in two” hells, we can first consider the combination of selecting two hells from seven possible hells, represented by C(7, 2) = 21. Within these seven hells, as three people enter two different places, and we do not consider the order of the three people, there are only two possible scenarios—either the first hell has two individuals and the second has one, or the first hell has one and the second has two. Therefore, the total is 21 × 2 = 42 forms. This can also be seen as a permutation calculation that does not consider the order of the three people but considers the order of the two hells, as it involves distributing three people, which is P(7, 2) = 42. When three people enter three different hells, this is calculated as C(7, 3) = 35.

Additionally, instances where two souls enter into the seven hells are mentioned:

Together, they may be born in Ratnaprabhā hell, or any other, till the lowest seventh, (7 forms) or, one in Ratnaprabhā and another in one of the six hells, (6 forms) or, one in śarkaraprabhā and another in one of the five, (5 forms) or, one in Vālukāprabhā and another in one of the four, (4 forms) or, one in Paṅkaprabhā and another in one of the three, (3 forms) or, one in Dhūmaprabhā and another in either of the two, (2 forms) or, one in Tamahprabhā and another in the lowest one, (1 form), total 28 forms.(Ibid, p. 36).

In simplified numerics, the combinations are:

{1, 1}, {2, 2}, {3, 3},…,{7, 7}, 7 forms

{1, 2}, {1, 3}, {1, 4},…,{1, 7}, 6 forms

{2, 3}, {2, 4},…,{2, 7} 5 forms

{3, 4},…,{3, 7}, 4 forms

And continuing until {6, 7}, making for a total of 28 combinations.

This case is equally applicable to the sum of an arithmetic series formula used in both Shisong lü and Samantapāsādikā, where S = n(n + 1)/2, for n = 7, yielding S = 7 × 8/2 = 28.

Overall, Hindu and Jain scriptures have furnished us with methods for calculating permutations and combinations, traceable even to before the Common Era. These observations suggest that during the era of the Theravāda-vinaya, ancient India might already have possessed a mature knowledge of how to apply permutations and combinations. However, we have not found any clues about the concept of oblong numbers.

Interestingly, we encounter this sequence in the Bakhshālī manuscript within a calculation concerning the impurities of gold:

Example (3 for Sūtra 27).

Listen to me. There are (nine) gold pieces, the quantities of which are (severally) one, two, three, four, five, six, seven, eight, and nine suvarnas. (Their) impurities begin with two māsas and decrease one by one in order. When you have mixed up those gold pieces into one, let (the impurity of the alloy) be told……

1	2	3	4	5	6	7	8	9
−2	−3	−4	−5	−6	−7	−8	−9	−10

‘Having multiplied by the impurities’ (Sütra 27). The result is: 2, 6, 12, 20, 30, 42, 56, 72, and 90. The sum of these is 330…….(Hayashi 1995, p.3 12, the sanskrit text see p. 205).

Although this example is unrelated to permutations and combinations, it provides us with a sequence of oblong numbers similar to those found in the Theravāda-vinaya.

The Bakhshālī manuscript primarily consists of “the original rules and examples” and “the commentary”, with the commentary dating from around the 7th century, indicating that the content of the original rules and examples significantly predates the 7th century.

While this case does not allow us to fully resolve the origin of the oblong number sequence used in the Theravāda-vinaya, it does demonstrate that India had a certain awareness of this sequence and utilized it in various contexts. This further verifies that the compilers of the Theravāda-vinaya did not randomly use permutation formulas, but rather based it on solid knowledge, setting out permutations as the selection of two elements from differing numbers in sets to arrive at the oblong number sequence. So, where might the oblong numbers used in the Theravāda-vinaya originate? It seems that we may find some clues in ancient Greek mathematics.

6. The Figurate Number in Ancient Greek

The oblong number, also known as the pronic number, is typically grouped with triangular numbers, polygonal numbers, and square numbers under the category of figurate numbers. Some scholars attribute the origin of figurate numbers to Pythagoras (c. 570–495 BCE) or the Pythagorean school. This view is supported by scholars such as Burnet (1914, pp. 52–54), Dickson (1952, p. 1), Zhmud (1989, pp. 261–62), Heath (1921, pp. 82–84), and D’Ooge et al. (1926, p. 254, note 3), among others.

The arguments of these scholars are primarily based on the following evidence:

(1)

Aristotle’s Metaphysics 1092b (Aristotle 1999, p. 293) comments that Eurytus, a member of the Pythagorean school and a disciple of Philolaus, represented numbers in the shapes of triangles and squares.

(2)

We can also consider references from Callimachus (third century BCE) or Speusippus (Zhmud 1989, pp. 261–62).

(3)

The most significant evidence comes from Nicomachus (c. 60—c. 120 AD) in his Introduction to Arithmetic, where he explicitly mentioned that the Pythagorean school employed oblong numbers and introduced a specific type of oblong number, known as heteromecic numbers, as follows:

Heteromecic numbers: 2, 6, 12, 20, 30, 42, and so on.

Oblong numbers: 8, 18, 32, 50, 72, 98 and so on.17

However, W.R. Knorr presents a different perspective, arguing that figurate numbers did not originate with Pythagoras. He divides the development of figurate numbers into four stages:

(1) The first mathematical appearance of figured numbers arose through the arrangement of pebble-units as squares and rectangles to illustrate the operation of multiplication of integers; (2) The study of these configurations, in the light of practices already familiar in the decorative arts would lead to the mathematical analysis of other patterns, in particular triangular arrays. From this came the discovery of the summation-generation of square and oblong numbers and the formulation of the concept of ‘gnomon’ as the foundation of the further study of figured plane and solid numbers. (3) During Plato’s lifetime, mathematicians like Theaetetus formalized parts of the theory of numbers by means of a modified representation of number, that is, the geometric representation by continuous quantities, lines, plane figures, and solids, rather than by discrete arrays. However, substantial portions of the older arithmetic, overlaid with Pythagorean and Platonic metaphysical speculations on the power and significance of numbers, were continued in a separate tradition of treatises by such authors as Philolaus, Speusippus, and Hypsicles, before definitive compilations were made by Nicomachus and the later neo-Pythagoreans (Knorr 1975, p. 145).

In conclusion, the true origin of figurate numbers or oblong numbers remains uncertain. However, we cannot overlook the consistency between the oblong numbers in ancient Greek mathematics and those found in the Theravāda-vinaya. This consistency suggests a possible connection, wherein the oblong number sequence in the Theravāda-vinaya, derived from calculating the permutations of two elements within different sets, might be a result of cultural exchanges between ancient Greece and ancient India.

7. Concluding Remarks

Within the context of Indian religions, Jainism has long been recognized for its extensive use of permutations and combinations. However, the application of these principles within Buddhist scriptures has received relatively little scholarly attention. This article aimed to address this gap by presenting a novel case study that illustrates the specific application of permutations and combinations in the Buddhist Vinaya.

This paper analyzes the use of cases of element arrangements involving r-root cases as found in the Theravāda-vinaya, revealing some correspondence with permutation formulas in modern mathematics. When combined with the r-root cases in the Theravāda-vinaya, a general permutation formula, P(n − r + 1, 2), can be derived. It can be said that the Theravāda-vinaya employs r-roots applications to continuously vary the total number of sets, thereby forming permutation calculations for selecting two elements from sets of different sizes. I propose that the purpose of such calculations in the Theravāda-vinaya is to manifest the characteristics of the oblong number sequence.

However, although the extensive use of permutations and combinations in Vedic and Jainism illustrates that ancient civilizations in India were familiar with these concepts—providing fertile ground for the permutation examples in the Theravāda-vinaya—we have not found instances in ancient Indian texts that predate the Theravāda-vinaya where the use of oblong numbers is evident in the way that it is in these examples. A mention of oblong numbers is found later in the Bakhshālī manuscript, regarding calculations for impurities in gold, and showcases practical and secular applications—a shift from the traditionally religious beginnings of Indian mathematics.

Additionally, traces of oblong numbers can also be found in Greek mathematics. While we cannot ascertain their exact origin, this commonality between ancient Greek mathematics and the Theravāda-vinaya may suggest that it is possibly the result of cultural exchange between ancient Greece and ancient India.

Moreover, this article underscores the diversity inherent in the study of the Buddhist Vinaya literature. It is conceivable that the compilers of the Theravāda-vinaya possessed relatively advanced mathematical knowledge, and the permutations and combinations presented in different texts exhibit variations, suggesting that each Vinaya has undergone a distinct developmental process. Although the current analysis has not yielded more specific insights into the development of Vinaya literature, it is anticipated that future diversified research in this area will uncover additional relevant information.