Permutations with a Single Cycle

UNICYCLE is a FORTRAN90 library which carries out some operations on permutations with a single cycle.

A permutation with a single cycle is a permutation P of N objects with the property that, for every object, it takes exactly N applications of P to restore an object to its original value.

Another way to think of this is that a permutation with a single cycle can be symbolized by a bracelet with N beads; the action of the permutation is to rotate the bracelet one position.

A permutation with a single cycle can be written in "sequence" form. Assuming the objects are labeled 1 through N, we start with 1, followed by P(1), followed by P(P(1)), and so on. Thus, the sequence (1,4,2,5,3) indicates the permutation which maps 1->4, 2->5, 3->1, 4->2, and 5->3.

A permutation with a single cycle is sometimes called a "cyclic permutation", but this term is also used with other meanings. Hence, we will affectionately call these objects "unicycles".


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


UNICYCLE is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

COMBO, a FORTRAN90 library which includes many combinatorial routines.

SELECT, a FORTRAN90 library which generates various combinatorial objects.

SUBSET, a FORTRAN90 library which generates, ranks and unranks various combinatorial objects.

Source Code:

Examples and Tests:

List of Routines:

You can go up one level to the FORTRAN90 source codes.

Last revised on 14 June 2012.