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Groups and combinatory counting

NikolinaMarković
Regional Talent Center Belgrade II, Belgrade, nikolinam98@gmail.com

1. Introduction B(G)=  f ( )

 G
Many counting problems, especially those which
include “similar” objects, would be hardly We define the cycle index polynomial as follows.
represented without the application of the group The polynominal
theory. This paper elaborates the counting of

nonequivalent configurations with respect to the Z G (z 1 , z 2 ,...,z n ) =  z 1 1 c ...z n n c
given permutation group. The largest contribution in  G
finding the methods for solving these problems was with variables z 1, z 2,...z n is called the Cycle index
made by a Hungarian mathematician George Pólya
and the most important result is certainly the Polya's polynomial of the observed group G.
Theorem (1937). The counting technique based on When there is a Cycle index polynomial, it is easy to
Polya's Theorem has a very significant application derive the Polya’sTheorem, shown by the following
in counting the number of some figures’ mapping,
counting the Boolean Counting Functions, graph formula:
counting, in chemistry (counting isomers), in F G (x 1 , x 2 ,...)= Z G (x 1 + x 2 + ..., + +..., +...).

statistical physics or crystallography.

2. Work methods The final section of this paper discusses thePolya's
Theorem application.
In order to apply the counting technique based on
the Polya's Theorem, we analyze the permutation 4. Conclusion
groups, Burnside’s lemma and cycle index
polynomial. For the analysis and solving the Based on the presentation made in this paper, we
counting problem we use two crafted cubes (Figure
1), which will hlep us to find a solution to this can conclude that the application of Burnside
problem. Lemma, generally has a visible defect. This
imperfection can be corrected to a great extent by

the Polya's Theorem. This theorem accounts for an
example how a mathematical method from the
algebra can be efficiently used in solving the
problems in combinatorics.

5. References

Figure 1 1. Duško Jojić, “Elements of Enumerative

3. Research results Combinatorics “
2. Dragan Cvetković ,PhD and Slobodan
Merging of the Burnside's lemma and the counting Simić , PhD, “Combinatorics and graphs”
technique based on function of generatrix has been
performed in the Polya'sTheorem. Burnside's lemma 3. Pavle Mladenović, “Combinatorics-
gives a formula for the number of orbits of a set X Material for young mathematicians”
under the influence of a permutation group G.