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GRUPE I KOMBINATORNA PREBROJAVANJA
GROUPS AND COMBINATORIAL COUNTING

Uĉenik : NIKOLINA MARKOVIĆ II razred Gimnazija u Lazarevcu
Mentor : VESNA RAJŠIĆ, profesor matematike Elektrotehniĉka škola „Nikola Tesla”
REZIME

Prebrojavanje predstavlja vaţan deo kombinatorike koji se bavi prebrojavanjem skupa

objekata sa odreĊenim svojstvima. Mnogi problemi prebrojavanja, a naroĉito oni kod kojih

meĊu posmatranim objektima ima “sliĉnih“, teško bi se mogli zamisliti bez primene teorije

grupa. Centralni rezultat svakako je teorema koju je dao Dţ. Polja, koja je jedan od

najelegantnijih rezultata u kombinatorici.

Tehnika prebrojavanja na bazi Poljine teoreme ima veoma znaĉajne primene u prebrojavanju

broja bojenja nekih figura, prebrojavanju Bulovih funkcija, prebrojavanju grafova, u hemiji,

u statistiĉkoj fizici.

U ovom radu ćemo se baviti analizom Poljine teoreme i njenom primenom.

Kljuĉne reĉi: teorija grupa, permutacije, funkcija generatrise, broj orbita i polinom ciklusnog

indeksa.

SUMMARY

Counting is an important part of the combinatory which deals with the counting of a set of

objects with certain properties. Many counting problems, especially those where among

observed objects there are “similar” ones, would hardly be imagined without application of

the group theory. The central result is certainly a theorem given by G. Polya, which is one of

the most elegant results in combinatory.

Technique of counting based on the Polya’s theorem has rather important applications in

counting the number of dyeing of some figures, the counting of Boolean’s functions, the

counting of graphs, in chemistry, in statistic physics.

In this paper, we will deal with the analysis of Polya’s theorem and its application.

Key words: group theory, permutations, generating function, number of orbits and

polynomial cyclical index.