Polya by Examples

1 Poly a by Examples KUNG-WEI YANG E-mail: kung-wei.y ang@wmich.edu Polya ’s Enumeration Theorem is one of the m ost useful tools dealing with the enumeration of patterns that are sy mmetric in som e way s. What follows is a procedure for obtaining the results of Polya' s Theorem directly, by passing the usual preliminaries ("cy cle index,"...). It is for people who want to use (and to understand) Polya' s Theorem. There is no proof here. There is no fancy ex ample here. “Alway s begin with t he simplest examples.” (Hilb ert) We will count t he number of distinct color patterns of the unit dis c in the plane partitioned into four quadrants (1, 2, 3, 4) by the two coordinate ax es. Each quadrant is to be colored in one of three colors: r ed, w hite, or b lu e. To highlight the different sy mmetry assumptions one m ay impose on the disc and the group action appropriate for eac h situatio n (and to help understand the meaning of the Theorem), we will do the counting in three different settings. Setting I . The disc is im movable. I n this case, no questio n of sy mm etry is i nvolved. Each quadrant can be colored in 3 way s. There are 4 quadra nts. The total nu mber of color patterns is 3 4 . Setting R . The disc can be freely rotated about i ts center. I n this si tuation, for example, the coloring red in the first quadrant white in t he three remaining quadrants, rwww, is indistinguishable from the colorings wrww, wwrw, and wwwr. (These last three colorings can be obtained from the first one by a rotatio n of 90 o , 2 180 o and 270 o , respectively . ) To find the correct number of distinct color patterns, all you need to do is to carry through the following simple procedure. Take the cy clic permutati on group of order 4 g enerated by the "90 0 rotation" (1234). Express each element of the group in its disj oint cy cle form. Do not omit 1-cy cles. C = {(1)(2)(3)(4), (1234), (13)(24), (1432)}. Change every digit to x and simplify. Ke ep all parentheses. Then take their average (add up the monomials and di vide the sum by the order of the group, 4). (1/4)((x) 4 + (x 2 ) 2 + 2(x 4 )). Substitut e (r i + w i + b i ) for x i . You will get I C (r, w, b) = (1/4)((r+w+b) 4 + (r 2 +w 2 +b 2 ) 2 + 2(r 4 +w 4 +b 4 )) = r 4 + r 3 w + br 3 + 2r 2 w 2 + 3br 2 w + 2b 2 r 2 + rw 3 + 3brw 2 + 3b 2 rw + b 3 r + w 4 + bw 3 + 2b 2 w 2 + b 3 w + b 4 . The coeffi cient of r i w j b k in I C (r, w, b) is the n umber of distinct color patterns with i q uadrants colored red, j qu adrants colored whi te, and k quad rants colored blue. The total nu mber of distinct color patterns is therefore = I C (1, 1, 1) = 24. Needless to say , there is much more information contained i n I C (r, w, b). Setting RF . The disc can be rotated and in addition it can be flipped over and the color is visible on bo th sides (same as a necklace with 4 beads colored red, white, blue). The only thing we have to do to accommodate the additional s y mmetry is to change the permutation group. We use, in place of the cyc lic permutation group of order 4 in Setting R, the dihedral permutation group of order 8 generated by the "90 0 rotation" (1234) and the "flip about the x-ax is" (14)(23). Every thing else works exactly the same way . Disjoint cycles: D = {(1)(2) (3)(4), (1234), (13)(24), (1432), (14)(23), (1)(3)(24), (12)(34), (13)(2)(4)}. Monomial average: (1/8)((x) 4 + 3(x 2 ) 2 + 2(x) 2 (x 2 ) + 2(x 4 )). Substitut ion (r i + w i + b i )  x i : I D (r, w, b) = (1/8)((r+w+b) 4 + 3(r 2 +w 2 +b 2 ) 2 + 2(r+w+b) 2 (r 2 +w 2 +b 2 ) + 2(r 4 +w 4 +b 4 )). 3 The total number of distin ct color patterns is = I D (1, 1, 1) = 21. Pick up a book on combi natorics (of some depth). Chances are good y ou will find in it a chapter devoted to Polya' s theory ([1] , [2], [3] , [4], [ 6]). An English translation of Polya 's original paper is in [5]. I thank Rena Yang for valuable suggestions. REFERENCES [1] C . Berg e, Principles of Comb inatorics , Academic Press, New York, 1971. [2] K. P . Bogar t, Introductory C ombinatorics, Harcourt/Academic Press, San Diego, 2000. [3] N. G. deBruijn, P oly a's theory of counting, Applied Combi natorial Mathematics (E. F. Beckenbach, ed.), Wiley , New York, 1964. [4] G. E. M artin, Counting: the Art of Enumerative Combinatorics, Sprin ger , New York, 2001. [5] G. P oly a and R . C. Read, Combinatorial Enumeration of Groups, Graph s and Chemical Compounds, S pringer-Ver lag, New York, 1987 [6] A. Tucker, Appli ed Combinatorics, J ohn Wiley & Sons, Hoboken, N. J . 2002.