Linkage

A permutation generation algorithm in the work of 13th-century Kabbalist Abraham Abulafia (\(\mathbb{M}\), via). The resulting permutation sequence is the one you get by reversing suffixes whose lengths form the sequence \((((2, 3)^2, 2, 4)^3, 2, 5)^4, \dots\) but that’s not the generation rule. Instead the rule is: to generate the permutations of \(1, 2, 3,\dots, n,\) form its \(n\) cyclically rotated permutations (starting with \(1, 2, 3,\dots, n,\)) and for each one in order, recursively generate the permutations of its length-\((n-1)\) suffix. The link claims that this is the first method for systematically listing permutations (long before the discovery of the Steinhaus–Johnson–Trotter algorithm for generating a different ordering, by 17th-century bell ringers). However, I’m not entirely convinced: it’s not the same as an enumeration algorithm, but factorials and lists of permutations were known long before to the Jain mathematicians, to scholars of the Talmudic period, and to…