The significance of the cycle index (polynomial) of symmetry group is deeply rooted in counting the number of configurations of an object excluding those that are symmetric (in terms of permutations). For example, the following problem can be solved as a direct application of the cycle index polynomial of the symmetry group. Note: I came across this problem as a Google's foo.bar challenge at Level 5 and solved it using a purely Group Theoretic approach. :) ----- Problem: Given positive integers w, h, and s, compute the number of distinct 2D grids of dimensions w x h that contain entries from {0, 1, ..., s-1}. Note that two grids are defined to be equivalent if one can be obtained from the other by switching rows and columns some number of times. ----- Approach: Compute the cycle index (polynomials) of S_w, and S_h, i.e. the Symmetry group on w and h symbols respectively. Compute the product of the two cycle indices while combining two monomials in such a way that for any pair of cycles c1, and c2 in the elements of S_w X S_h, the resultant monomial contains terms of the form: $$ x_{lcm(|c1|, |c2|)}^{gcd(|c1|, |c2|)} $$ Return the specialization of the product of cycle indices.
The significance of the cycle index (polynomial) of symmetry group
is deeply rooted in counting the number of configurations
of an object excluding those that are symmetric (in terms of permutations).
For example, the following problem can be solved as a direct
application of the cycle index polynomial of the symmetry
group.
Note: I came across this problem as a Google's foo.bar challenge at Level 5
and solved it using a purely Group Theoretic approach. :)
-----
Problem:
Given positive integers
w, h, and s,
compute the number of distinct 2D
grids of dimensions w x h that contain
entries from {0, 1, ..., s-1}.
Note that two grids are defined
to be equivalent if one can be
obtained from the other by
switching rows and columns
some number of times.
-----
Approach:
Compute the cycle index (polynomials)
of S_w, and S_h, i.e. the Symmetry
group on w and h symbols respectively.
Compute the product of the two
cycle indices while combining two
monomials in such a way that
for any pair of cycles c1, and c2
in the elements of S_w X S_h,
the resultant monomial contains
terms of the form:
$$ x_{lcm(|c1|, |c2|)}^{gcd(|c1|, |c2|)} $$
Return the specialization of
the product of cycle indices.
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