Counting Symmetries with Burnside's Lemma and Polya's Theorem

Md Taufiq Nasseef

Abstract

Counting concerns a large part of combinational analysis. Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy-Frobenius lemma or the orbit-counting theorem, is often useful in taking account of symmetry when counting mathematical ob-jects. The Polya's theorem is also known as the Redeld-Polya Theorem which both follows and ultimately generalizes Burnside's lemma on the number of orbits of a group action on a set. Polya's Theory is a spectacular tool that allows us to count the number of distinctitems given a certain number of colors or other characteristics. Sometimes it is interesting to know more information about the characteristics of these distinct objects. Polya's Theory isa unique and useful theory which acts as a picture function by producing a polynomial that demonstrates what the dierent congurations are, and how many of each exist.The dynamics of counting symmetries are the most interesting part. This subject has been a fascination for mathematicians and scientist for ages. Here 16 Bead Necklace, patterns of n tetrahedron with 2 colors, patterns of n cubes with 3 and 4 colorings and so on have beensolved.

Keywords

Group Action, Burnside's Lemma, Polya's Fundamental Theorem

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