Polya Enumeration

1. Consider the molecules which can be formed from one carbon atom C

linked to four radicals, each of which can be HOCH2 (hydroxymethyl),C2H5 (ethyl), Cl (chlorine), H (hydrogen). Picture this situation withC in the centre of a regular tetrahedron and the radicals occupying thecorners.

(a) Show that there are 36 possible molecules;

(b) Show that there are 11 molecules which contain exactly one H

radical;

(c) Compute the generating function

H(x) = h0 + h1x + h2x2 + h3x

3 + h4x4

where hi is the number of molecules containing exactly i H radi-cals.

2. Show that the group of symmetries of the cube, regarded as a group ofpermutations of the faces, has cycle index

1

24(x6

1 + 6x21x4 + 3x2

1x22 + 8x2

3 + 6x32).

3. Write down the cycle index of S4. Show that the cycle index of S(2)4

(the action induced on unordered pairs) is

1

24(x6

1 + 9x21x

22 + 8x2

3 + 6x12x

14).

Hence obtain the number of nonisomorphic graphs on four vertices andm edges, for 0 ≤ m ≤ 6.

Find the cycle index of S[2]4 (the action induced on ordered pairs), and

hence find the number of non-isomorphic digraphs on four vertices.

4. Past papers:

(a) June 81 Nos 4 & 3

(b) Dec 91 No 2

(c) June 93 No 8

(d) June 95 Nos 5 & 6

(e) June 96 No 3

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(f) May 97II No 2 (Sect B)

(g) June 99I No 2 (Sect A)

(h) June 99II No 1 (Sect A)

(i) May 97I No 2 (Sect B)

(j) June 2008 No 4

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