Math 146: Algebraic CombinatoricsHomework 8This problem set is due Friday, May 21. This is a half problem set.Do the following problems:GK8.1.Analternating permutationis a permutationσof[n]such thatσ(k+1)-σ(k)is positive whenkis odd and negative whenkis even. For example, 1423 is an alternating permutation. Letanbe thenumber of alternating permutations of[n], and letA(x)be their exponential generating function.(a)Letbnbe the set of “zigzag permutations”, which are those that are either alternating or satisfythe reverse condition, thatσ(k+1)-σ(k)is positive whenkis even and negative whenkisodd. Show thatbn=2anand thusB(x) =2A(x)for their generating functions. (We setb0=2by default.)*(b)Prove the “two animal” relationB0(x) =A(x)2+1.(c)Use (a) and (b) and the initial conditionA(0) =a0=1 to establish thatA(x) = (secx)+(tanx).GK8.2.The version of Abel’s theorem that I gave in class, although correct, is not the strongest one avail-able for our purposes. It is a theorem that iff(x)
This is the end of the preview.
access the rest of the document.