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Hints and Comments for Some Exercises Hints and Comments for Some Exercises Chapter 1 1.5 Consider A(Hn)2 and use Exercise 1.3. V ,V ,...,V 1.6 (a) First count the number of sequences i0 i1 i for which there exists a closed walk with vertices v0,v1,...,v = v0 (in that order) such that vj ∈ Vij . 1.11 Consider the rank of A(), and also consider A()2. The answer is very simple and does not involve irrational numbers. 1.12 (b) Consider A(G)2. Chapter 2 2.2 Give an argument analogous to the proof of Theorem 8.8.9. 2.3 This result is known as the “middle levels conjecture” and was proved by Torsten Mütze in 2016. 2.5 (c) Mimic the proof for the graph Cn, using the definition χu,χv= χu(w)χv(w), w∈Zn where an overhead bar denotes complex conjugation. Chapter 3 3.4 You may find Example 3.1 useful. 3.8 It is easier not to use linear algebra. 3.9 See previous hint. © Springer International Publishing AG, part of Springer Nature 2018 245 R. P. Stanley, Algebraic Combinatorics, Undergraduate Texts in Mathematics, https://doi.org/10.1007/978-3-319-77173-1 246 Hints and Comments for Some Exercises 3.11 First show (easy) that if we start at a vertex v and take n steps (using our random walk model), then the probability that we traverse a fixed closed walk W is equal to the probability that we traverse W in reverse order. 3.13 See hint for Exercise 3.8. Chapter 4 4.4 (b) One way to do this is to count in two ways the number of k-tuples n (v1,...,vk) of linearly independent elements from Fq : (1) first choose v1, then v2, etc., and (2) first choose the subspace W spanned by v1,...,vk, and then choose v1, v2,etc. 4.4 (c) The easiest way is to use (b). 4.7 (b) This result was proved by R. P. Dilworth in 1950 and is known as Dilworth’s theorem. Chapter 5 ∼ 5.5 (a) Show that Nn = Bn/G for a suitable group G. n = p 5.9 (a) Use Corollary 2.4 with 2 . 5.13 Use Exercise 5.12. Chapter 6 6.2 (b) Not really a hint, but the result is equivalent [why?] to the case r = m, s = n, t = 2, and x = 1 of Exercise 36. 6.3 Consider μ = (8, 8, 4, 4). 6.5 First consider the case where S has ζ elements equal to 0 (so ζ = 0or1),ν elements that are negative, and π elements that are positive, so ν + ζ + π = 2m + 1. Chapter 7 7.3 Hint. Dark red and burgundy are the same color! For those who are interested, the cycle indicator ZG of the full symmetry group G of the dodecahedron acting on the vertices is 1 z20 + 15z10 + 20z2z6 + 24z4 + 15z4z8 + z10 + 20z z3 + 24z2 . 120 1 2 1 3 5 1 2 2 2 6 10 The number of inequivalent vertex colorings using two distinct colors is ZG(2, 2,...,2) = 9436. 7.19 (a) Use Pólya’s theorem. Hints and Comments for Some Exercises 247 Chapter 8 8.3 Encode a maximal chain by an object that we already know how to enumerate. 8.7 Partially order by diagram inclusion the set of all partitions whose diagrams can be covered by nonoverlapping dominos, thereby obtaining a subposet Y2 ∼ of Young’s lattice Y . Show that Y2 = Y × Y . 8.14 Use induction on n. Z (z ,z ,...) 8.17 (a) One way to do this is to use the generating function n≥0 Sn 1 2 n x for the cycle indicator of Sn (Theorem 7.13). Another method is to find a recurrence for B(n + 1) in terms of B(0), . , B(n) and then convert this recurrence into a generating function 8.18 Consider the generating function tkqn G(q, t) = κ(n → n + k → n) (k!)2 k,n≥0 and use (8.25). 8.20 (b) Consider the square of the adjacency matrix of Yj−1,j . 8.24 Use Exercise 8.14. 8.33 Use an inclusion–exclusion argument (a topic not covered in the text). Chapter 9 p− 9.1 There is a simple proof based on the formula κ(Kp) = p 2, avoiding the Matrix-Tree Theorem. 9.3 Consider the matrices L − nI and L − (n − m)I. 9.2 (c) Use the fact that the rows of L sum to 0, and compute the trace of L. 9.7 (b) Use Exercise 1.3. 9.8 (a) For the most elegant proof, use the fact that commuting p × p matrices A and B can be simultaneously triangularized, i.e., there exists an invertible matrix X such that both XAX−1 and XBX−1 are upper triangular. 9.8 (d) Use Exercise 9.10(a). 9.9 Let G∗ be the full dual graph of G, i.e., the vertices of G∗ are the faces of G, including the outside face. For every edge e of G separating two faces R and S of G, there is an edge e∗ of G∗ connecting the vertices R and S. Thus G∗ will have some multiple edges, and #E(G) = #E(G∗). First show combinatorially that κ(G) = κ(G∗). (See Corollary 11.19.) 9.10 (a) Use the Binet–Cauchy theorem. 9.12 (a) The Laplacian matrix L = L(G) acts on the space RV(G), the real vector space with basis V(G). Consider the subspace W of RV(G)spanned by the elements v + ϕ(v), v ∈ V(G). 9.13 (a) Let s(n,q,r) be the number of n × n symmetric matrices of rank r over Fq . Find a recurrence satisfied by s(n,q,r), and verify that this recurrence is satisfied by 248 Hints and Comments for Some Exercises ⎧ ⎪ t 2i 2 t−1 ⎪ q n−i ⎪ · (q − 1), 0 ≤ r = 2t ≤ n ⎨ q2i − 1 s(n,q,r) = i=1 i=0 ⎪ t q2i 2t ⎪ n−i ⎩⎪ · (q − 1), 0 ≤ r = 2t + 1 ≤ n. q2i − 1 i=1 i=0 9.14 Any of the three proofs of the Appendix to Chapter 9 can be carried over to the present exercise. Chapter 10 10.4 (a) Use a suitable generalization of the deBruijn graph. (b) Use techniques similar to those in Theorem 7.7.10. 10.5 (b) Use the Perron–Frobenius theorem (Theorem 3.3). 10.8 (a) Consider A. 10.8 (f) There is an example with nine vertices that is not a de Bruijn graph. 10.8 (c) Let E be the (column) eigenvector of A(D) corresponding to the largest eigenvalue. Consider AE and At E, where t denotes transpose. Chapter 11 11.2(d) Let e be an edge of G that is not an isthmus. Let G − e denote G with the edge e removed, and let G/e denote G with the edge e contracted to a point (so G/e has one less vertex than G). Find a simple recurrence satisfied by CG(n) in terms of CG−e(n) and CG/e(n), and then use induction. 11.5 Use the unimodularity of the basis matrices CT and BT . 11.6 Use the formula (11.5) for the resistance of a parallel connection. 11.8 (a) Mimic the proof of Theorem 9.8 (the Matrix-Tree Theorem). 11.8 (b) Consider ZZt . Chapter 12 12.9. Compute the reduced Euler characteristic of X and consider the last step of a possible shelling. 12.10. Modify the dunce hat of the previous exercise. 12.12. There are two examples with f -vector (6, 14, 9) and none smaller. 12.13. (b) Consider the hint for Problem 12.12. (c) Use Exercise 12.10. 12.18. The following property of triangulations of spheres is sufficient for solving this exercise. For every F ∈ we have dim lk (F ) χ(˜ lk (F )) = (−1) . Hints and Comments for Some Exercises 249 Here χ˜ denotes reduced Euler characteristic, and lk denotes link, as defined in Remark 12.26. 12.20. There are five minimal nonfaces. 12.21. One approach is to consider minimal ideals I of K[], say, for which K[]/I is a polynomial ring, i.e., isomorphic to K[y1,...,yr ] for inde- terminates y1,...,yr . 12.30. Answer: 588. Without shellability, the answer is 589. 12.31. Let I be the ideal of K[ ] generated by all monomials xF , where F ∈ . Clearly K[ ] is isomorphic to K[ ]/I (as a K-algebra). Let θ1,...,θd ∈ K[ ]1 satisfy Property (P), and consider the natural map f : K[ ]/(θ1,...,θd ) → (K[ ]/I)/(θ1,...,θd ). θ ∈ K[ ] θ = x 12.32. Consider i 1 defined by i xj ∈Vi j . Also consider the product xj θi in the ring K[ ]/(θ1,...,θd ) for all 1 ≤ i ≤ d and xj ∈ Vi. Chapter 13 13.4 The best strategy involves the concept of odd and even permutations. 13.5 For the easiest solution, don’t use linear algebra but rather use the original Oddtown theorem. 13.12 What are the eigenvalues of skew-symmetric matrices? 13.15 Consider the incidence matrix M of the sets and their elements. Consider two cases: det M = 0 and det M = 0. 13.19 Consider the first three rows of H . Another method is to use row operations to factor a large power of 2 from the determinant. 13.22 It is easiest to proceed directly and not use the proof of Theorem 13.18.
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