History of combinatorics

The mathematical field of

Earliest records

The earliest recorded use of combinatorial techniques comes from problem 79 of the

In Greece, Plutarch wrote that Xenocrates of Chalcedon (396–314 BC) discovered the number of different syllables possible in the Greek language. This would have been the first attempt on record to solve a difficult problem in permutations and combinations.^[2] The claim, however, is implausible: this is one of the few mentions of combinatorics in Greece, and the number they found, 1.002 × 10 ¹², seems too round to be more than a guess.^[3][4]

Later, an argument between Chrysippus (3rd century BCE) and Hipparchus (2nd century BCE) of a rather delicate enumerative problem, which was later shown to be related to Schröder–Hipparchus numbers, is mentioned.^[5][6] There is also evidence that in the Ostomachion, Archimedes (3rd century BCE) considered the configurations of a tiling puzzle,^[7] while some combinatorial interests may have been present in lost works of Apollonius.^[8][9]

In India, the Bhagavati Sutra had the first mention of a combinatorics problem; the problem asked how many possible combinations of tastes were possible from selecting tastes in ones, twos, threes, etc. from a selection of six different tastes (sweet, pungent, astringent, sour, salt, and bitter). The Bhagavati is also the first text to mention the choose function.^[10] In the second century BC, Pingala included an enumeration problem in the Chanda Sutra (also Chandahsutra) which asked how many ways a six-syllable meter could be made from short and long notes.^[11][12] Pingala found the number of meters that had ${\displaystyle n}$ long notes and ${\displaystyle k}$ short notes; this is equivalent to finding the

The ideas of the Bhagavati were generalized by the Indian mathematician

The ancient Chinese book of divination I Ching describes a hexagram as a permutation with repetitions of six lines where each line can be one of two states: solid or dashed. In describing hexagrams in this fashion they determine that there are ${\displaystyle 2^{6}=64}$ possible hexagrams. A Chinese monk also may have counted the number of configurations to a game similar to

Magic squares remained an interest of China, and they began to generalize their original ${\displaystyle 3\times 3}$ square between 900 and 1300 AD. China corresponded with the Middle East about this problem in the 13th century.

al-Khalil ibn Ahmad

who considered the possible arrangements of letters to form syllables. His calculations show an understanding of permutations and combinations. In a passage from the work of Arab mathematician Umar al-Khayyami that dates to around 1100, it is corroborated that the Hindus had knowledge of binomial coefficients, but also that their methods reached the middle east.