A 1677 bell-ringing manual worked out factorials before most mathematicians named them
Fabian Stedman wanted to know how many orders a set of church bells could ring in. Two bells give two sequences, three give six, and by five he had tabulated 120 before giving up on writing them all out. His 1677 reasoning was a recursive account of permutations, the mathematics of rearrangement.
A permutation is a one-to-one mapping of a set onto itself, or more intuitively a reordering of its members. The set of 1, 2 and 3 can be arranged in six ways, and an anagram of a word with no repeated letters is another example. The number of orderings of n distinct objects is n factorial, the product of every whole number from 1 to n. Choosing and ordering only k items from a larger set gives partial permutations, and all permutations of a set together form the symmetric group, whose operation is doing one rearrangement after another.
Uses stretch well beyond pure mathematics. Computer scientists analyse sorting algorithms with them, quantum physicists describe particle states, and biologists describe RNA sequences. Combinatorics and group theory treat permutations of finite sets as a core topic.
Arrangement puzzles are ancient. Chinese hexagrams in the I Ching, permutation-like objects, date back as early as 1000 BC. Plutarch credited Xenocrates of Chalcedon, who lived from 396 to 314 BC, with counting the possible syllables of Greek, the first recorded attack on a hard counting problem of this kind. The Arab scholar al-Khalil, who lived from 717 to 786, used permutations and combinations in his Book of Cryptographic Messages to list every possible Arabic word, with and without vowels. By about 1150 the rule for counting arrangements of n objects was known in India and appears in Bhāskara II's Lilavati. Stedman himself went on to count the ways of ordering letters of the alphabet, and 20 horses in a stable.
Around 1770 Joseph Louis Lagrange noticed that how the roots of a polynomial equation can be shuffled relates to whether the equation can be solved. Évariste Galois developed the idea into Galois theory, which settles exactly which single-variable polynomial equations can be solved using radicals, an early case of permutations illuminating seemingly unrelated problems.
Source: Permutation
