====== Permutation ====== ---- ====Basics==== * ''Permutation'' represents a mathematical permutation. * ''Permutation'' can be inputted in both //one-line// and //cycle// notation using ''.p'' property. * ''Permutation'' can can represent both permutational symmetry or antisymmetry. * ''[].p'' represents identity permutation. * Internally, Redberry stores ''Permutation'' in ''byte[]'', ''short[]'' or ''int[]'' array dynamically choosing the representation according to the degree of permutation. ====Examples==== Input ''Permutation'' in one-line or cycle notation: //permutation in one-line notation def p1 = [0, 2, 5, 6, 7, 1, 3, 4].p //same permutation in cycle notation def p2 = [[1, 2, 5], [4, 7], [3, 6]].p assert p1 == p2 ---- ''Permutation'' may represent permutational symmetry or antisymmetry; in order to convert symmetry to antisymmetry and vice versa one can use minus: //antisymmetry def asym = -[[0, 4, 2], [1, 3]].p //symmetry def sym = -asym One should be careful when inputting antisymmetries, since if a permutation order is odd (i.e. $p^r = 1$, where $p$ is a permutation and $r$ its order which is odd), then, obviously, such antisymmetry is inconsistent and Redberry will throw exception: def perm = [[0, 2, 5], [6, 7, 4]].p println perm.order() > 3 //this will throw exception println -perm > InconsistentGeneratorsException ---- One can apply permutation to some list using right shift operator: def p = [[0, 1], [2, 3]].p println p >> [10, 9, 8, 7] > [9, 10, 7, 8] println p >> ['a', 'b', 'c', 'd', 'e'] > [b, a, d, c, e] ---- The algebraic operations on permutations (composition, pow, inverse) can be performed in the following way: def p = [[0, 5, 4], [1, 3]].p //inverse println p**(-1) >[[0, 4, 5], [1, 3]] //p1 * p1 * p1 println p**(3) > [[1, 3]] //inverse of (p1 * p1) println p**(-2) > [[0, 5, 4]] def oth = [[0, 1], [2, 3]].p //apply oth after p println p * oth > [[0, 5, 4, 1, 2, 3]] //apply p after oth println oth * p > [[0, 3, 2, 1, 5, 4]] The convention on composition of permutations is the following: if ''a'' and ''b'' two permutations, then the result of applying composition ''a*b'' is equivalent to applying ''b'' after ''a''. ---- In order to obtain a new position of //i//-th element under permutation one can use ''[]'' operator: def p = [[0, 5, 4], [1, 3]].p assert p[0] == 5 assert p[4] == 0 ====Additional features==== The following table summarises some additional features of ''Permutation'':

`.degree()`	returns degree of permutation, i.e. largest moved point plus one.
`.order()`	calculates and returns the order of permutation.
`.parity()`	returns parity of permutation (`0` for even and `1` for odd).
`.antisymmetry()`	returns whether `Permutation` is antisymmetry.

More specialised features of ''Permutation'' can be found in API (see [[#See also| JavaDocs]]). =====See also===== * Related guides: [[documentation:guide:Permutations and permutation groups]] * Related reference material: [[documentation:ref:permutationgroup]] * JavaDocs: [[http://api.redberry.cc/redberry/1.1.9/java-api/cc/redberry/core/groups/permutations/Permutation.html| Permutation]] * Source code: [[https://bitbucket.org/redberry/redberry/src/tip/core/src/main/java/cc/redberry/core/groups/permutations/Permutation.java|Permutation.java]]