Combinatorics Through Guided Discovery, 2004a

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C.4. The Product Principle for EGFs 161 in Problems 373 and Problem 389. What is the EGF for the number p n of ways to paint n streetlight poles with some fixed number k of colors of paint. · Problem 402. Use the general product principle for EGFs or one of its corollaries to explain the relationship between the EGF for arranging books on one shelf and the EGF for arranging books on n shelves in Problem 383. ⇒ Problem 403. (Optional) Our very first example of exponential generating functions used the binomial theorem to show that the EGF for k-element permutations of an n element set is (1 + x) n . Use the EGF for k-element permutations of a one-element set and the product principle to prove the same thing. Hint: Review the alternate definition of a function in Section 3.1.2. (h) Problem 404. What is the EGF for the number of ways to paint n streetlight poles red, white blue, green and yellow, assuming an even number of poles must be painted green and an even number of poles must be painted yellow? Give a formula for the number of ways to paint n poles. (Don’t forget the factorial!) (h) ⇒ · Problem 405. What is the EGF for the number of functions from an n- element set onto a one-element set? (Can there be any functions from the empty set onto a one-element set?) What is the EGF for the number c n of functions from an n-element set onto a k element set (where k is fixed)? Use this EGF to find an explicit expression for the number of functions from a k-element set onto an n-element set and compare the result with what you got by inclusion and exclusion. ⇒ · Problem 406. In Problem 142 You showed that the Bell Numbers B n satisfy the equation B n+1 = ∑ n k=0 (n k )B n−k (or a similar equation for B n .) Multiply both sides of this equation by xn n! and sum from n =0to infinity. On the left hand side you have a derivative of a certain EGF we might call B(x). On the right hand side, you have a product of two EGFs, one of which is B(x). What is the other one? What differential equation involving B(x) does this give you. Solve the differential equation for B(x). This is the EGF for the Bell numbers!.
162 C. Exponential Generating Functions ⇒ Problem 407. Prove that n2 n−1 = ∑ n k=1 (n )k k by using EGFs. (h) · Problem 408. In light of Problem 382, why is the EGF for the Stirling numbers S(n, k) of the second kind not (e x − 1) n ? What is it equal to instead? C.5 The Exponential Formula Exponential generating functions turn out to be quite useful in advanced work in combinatorics. One reason why is that it is often possible to give a combinatorial interpretation to the composition of two exponential generating functions. In particular, if f (x) = ∑ n i=0 a i xi i! and g(x) = ∑ ∞ j=1 b j x j j! , it makes sense to form the composition f (g(x)) because in so doing we need add together only finitely many terms in order to find the coefficient of xn n! in f (g(x)) since in the EGF g(x) the dummy variable j starts at 1. Since our study of combinatorial structures has not been advanced enough to give us applications of a general formula for the composition of the EGF, we will not give here the combinatorial interpretation of this composition. However we have seen some examples where one particular composition can be applied. Namely, if f (x) = e x = (x), then f (g(x)) = exp(g(x)) is well defined when b 0 =0. We have seen three examples in which an EGF is e f (x) where f (x) is another EGF. There is a fourth example in which the exponential function is slightly hidden. · Problem 409. If f (x) is the EGF for the number of partitions of an n-set into one block, and g(x) is the EGF for the total number of partitions of an n-element set, that is, for the Bell numbers B n , how are the two generating functions related? · Problem 410. Let f (x) be the EGF for the number of permutations of an n-element set with one cycle of size one or two. What is f (x)? What is the EGF g(x) for the number of permutations of an n-element set all of whose cycles have size one or two, that is, the number of involutions in S n ?How are these two exponential generating functions related? Problem 411. ⇒ · Let f (x) be the EGF for the number of permutations of an n- element set that have exactly one two-cycle and no other cycles. Let g(x) be the EGF for the number of permutations which are products of two-cycles only, that is, for tennis pairings. (That is, they are not a product of two-cycles
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