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MATH 630–600. Enumerative Combinatorics
Assignment 7. Due on Wednesday, December 3, 2008
- Let n be a square-free integer, n = p 1 p 2 · · · pk with the pi distinct primes. Show that the maximum number of divisors of n which do not divide one another is
( (^) k bk/ 2 c
- Fix k ∈ N. For each n ∈ Z+, let σk(n) =
d|n
dk.
(a) Check that σ 0 (n) = ν(n) (the number of divisors of n) and σ 1 (n) = σ(n) (the sum of the divisors of n). (b) Let p 1 ,... , pr be the different prime divisors of n. Show that
nk^ = σk(n) −
i
σk( n pi
i<j
σk( n pipj
) + · · · + (−1)rσk( n p 1 · · · pr
(c) Verify this identity when n = 12 and k = 2.
- For each n ∈ Z+, let φ(n) := #{i ∈ [n] : gcd(i, n) = 1}. This is the Euler function. For each positive divisor d of n, let
Φ(d, n) := {i ∈ [n] : gcd(i, n) = d}.
(a) Show that there is a disjoint decomposition
[n] =
d|n
Φ(d, n),
and a bijection Φ(d, n) ∼= Φ(1, n/d). (b) Deduce that n =
d|n
φ(d).
(c) Deduce that φ(n) = n
p|n
p
where p ranges over prime factors of n.
- Given a subspace V of Fnq , let α(V ) be the number of subsets of V and β(V ) the number of spanning subsets of V. Here a subset W of V is spanning if V equals the vector space spanned by all the vectors in W. (On the other hand, the vectors in W are not necessarily independent.)
(a) Show that α(V ) = 2q dim(V ) and α(V ) − 1 =
U ≤V
β(U ).
(Note that the empty subset of V does not span a subspace. ) (b) Deduce that the number of spanning subsets of Fnq is
∑^ n
k=
n k
q
(−1)n−kq(
n−k 2 )(2qk − 1).
- Let S = {s 1 , s 2 ,... , } be a set of positive integers. Let hS (n) be the number of partitions of the set [n] into blocks so that each block size is an element of S. Let HS (x) be the exponential generating function of the sequence {hS (n)}. Prove that
HS (x) = exp(
i≥ 1
xsi si!
- An involution is a permutation π such that π^2 = id. Let i(n) be the number of involutions of length n. Compute the exponential generating function for the sequence {i(n)}.
- A threshold graph is a simple (i.e. no loops or multiple edges) graph which may be defined inductively as follows:
(a) The empty graph is a threshold graph. (b) If G is a threshold graph, then so is the disjoint union of G with a one-vertex graph. (c) If G is a threshold graph, then so is the (edge) complement of G.
Let t(n) be the number of threshold graphs with vertex set [n], with t(0) = 1, and let s(n) denote the number of such graphs with no isolated vertex, (so s(0) = 1, s(1) = 0). Set T (x) = Et(x) and S(x) = Es(x).
(a) List all threshold graph on [4], and compute t(n), s(n) for n = 2, 3 , 4. (b) Show that T (x) = exS(x), and T (x) = 2S(x) + x − 1.
(c) Deduce that
T (x) = ex(1 − x)/(2 − ex), S(x) = (1 − x)/(2 − ex).
- Find the unique power series F (x) such that for all n ∈ N, we have [xn]F (x)n+1^ = 1.
