permutations (version 1.0-5)

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)

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.

Value

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

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

Run this code
# NOT RUN {
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] 



# }

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