permutations (version 1.0-9)

Ops.permutation: Arithmetic Ops Group Methods for permutations

Description

Allows arithmetic operators to be used for manipulation of permutation objects such as addition, multiplication, division, integer powers, etc.

Usage

# S3 method for permutation
Ops(e1, e2)
cycle_power(x,pow)
cycle_power_single(x,pow)
cycle_sum(e1,e2)
cycle_sum_single(c1,c2)
group_action(e1,e2)
word_equal(e1,e2)
word_prod(e1,e2)
word_prod_single(e1,e2)
permprod(x)
vps(vec,pow)
ccps(n,pow)
helper(e1,e2)

Value

None of these functions are really intended for the end user: use the ops as shown in the examples section.

Arguments

x,e1,e2

Objects of class “permutation”

c1,c2

Objects of class cycle

pow

Integer vector of powers

vec

In function vps(), a vector of integers corresponding to a cycle

n

In function ccps(), the integer power to which cycle(seq_len(n)) is to be raised; may be positive or negative.

Author

Robin K. S. Hankin

Details

The function Ops.permutation() passes binary arithmetic operators (“+”, “*”, “/”, “^”, and “==”) to the appropriate specialist function.

Multiplication, as in a*b, is effectively word_prod(a,b); it coerces its arguments to word form (because a*b = b[a]).

Raising permutations to integer powers, as in a^n, is cycle_power(a,n); it coerces a to cycle form and returns a cycle. Negative and zero values of n operate as expected. Function cycle_power() is vectorized; it calls cycle_power_single(), which is not. This calls vps() (“Vector Power Single”), which checks for simple cases such as pow=0 or the identity permutation; and function vps() calls function ccps() which performs the actual number-theoretic manipulation to raise a cycle to a power.

Raising a permutation to the power of another permutation, as in a^b, is idiom for inverse(b)*a*b, sometimes known as group action; the notation is motivated by the identities x^(yz)=(x^y)^z and (xy)^z=x^z*y^z.

Permutation addition, as in a+b, is defined if the cycle representations of the addends are disjoint. The sum is defined as the permutation given by juxtaposing the cycles of a with those of b. Note that this operation is commutative. If a and b do not have disjoint cycle representations, an error is returned. This is useful if you want to guarantee that two permutations commute (NB: permutation a commutes with a^i for i any integer, and in particular a commutes with itself. But a+a returns an error: the operation checks for disjointness, not commutativity).

Permutation “division”, as in a/b, is a*inverse(b). Note that a/b*c is evaluated left to right so is equivalent to a*inverse(b)*c. See note.

Function helper() sorts out recycling for binary functions, the behaviour of which is inherited from cbind(), which also handles the names of the returned permutation.

Examples

x <- rperm(20,9) # word form
y <- rperm(20,9) # word form

x*y  # word form

x^5  # coerced to cycle form

x^as.cycle(1:5)  # group action; coerced to word.

x*inverse(x) == id  # all TRUE



# the 'sum' of two permutations is defined if their cycles are disjoint:
as.cycle(1:4) + as.cycle(7:9)

data(megaminx)
megaminx[1] + megaminx[7:12]