19 research outputs found
Permuting operations on strings and their relation to prime numbers
Some length-preserving operations on strings only permute the symbol positions in strings; such an operation gives rise to a family of similar permutations. We investigate the structure and the order of the cyclic group generated by . We call an integer -{\em prime} if consists of a single cycle of length (). Then we show some properties of these -primes, particularly, how -primes are related to -primes as well as to ordinary prime numbers. Here and range over well-known examples (reversal, cyclic shift, shuffle, twist) and some new ones based on Archimedes spiral and on the Josephus problem
Permuting operations on strings: Their permutations and their primes
We study some length-preserving operations on strings that permute the symbol positions in strings. These operations include some well-known examples (reversal, circular or cyclic shift, shuffle, twist, operations induced by the Josephus problem) and some new ones based on Archimedes spiral. Such a permuting operation gives rise to a family of similar permutations. We investigate the structure and the order of the cyclic group generated by such a permutation . We call an integer -prime if consists of a single cycle of length (). Then we show some properties of these -primes, particularly, how -primes are related to -primes as well as to ordinary prime numbers
Involution factorizations of Ewens random permutations
An involution is a bijection that is its own inverse. Given a permutation
of let denote the number of ways to
express as a composition of two involutions of The statistic
is asymptotically lognormal when the symmetric groups
are each equipped with Ewens Sampling Formula probability
measures of some fixed positive parameter This paper strengthens and
generalizes previously determined results on the limiting distribution of
for uniform random permutations, i.e. the specific case
of . We also investigate the first two moments of
itself.Comment: 23 pages, no figures. Some minor edits. Extra material added to
sections 2 and 4 and concluding remark