By Euler L.

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E. The number ofsequences (1)1,1>2, ••• ,l>n) ofO's and l's such that 1>1 ::s; 1>2? 1>3 ::s; 1>4 ? I> 5 ::s; .... f. L a 1 a 2 ••• ab where the sum is over a1l2 n - 1 compositions a 1 + a 2 + ... + ak = n. g. L (2 al - 1)'" (2 ak - 1), summed over the same set as (0. h. L2#{i:a;=1}, summed over the same set as (f). i. The number of sequences (15 1 ,15 2 " " , bn ) of O's, 1's, and 2's such that 0 is ne ver followed immediately by 1. [2 - ] [2] [2 - ] [2] [2] [2 + ] [2 + ] [2+] [2+] [2] 15. P. Find a simple expression involving Fibonacci numbers for the number of sequences (Tl' T2 , ••• , 7;,) of subsets T; of [k] such that Tl S; T2 ;;;2 T3 S;T4 ;;;2···· [2 +] 16.

Let 6 i (M)(1 :;;; i :;;; 2) consist of those nE 6(M) whose last element is i, and let MI = {lk-I,2n - k }, M2 = {l k ,2n - k - I }. IfnE6 1 (M) and n = 0"1, then 0" E 6 (MI ) and i(n) = n - k + i(O"). If nE 6 2 (M) and n = 1'2, then l' E 6(M2 ) and i(n) = i(1'). Hence Q(n, k) = L qi(<1 l +n-k + L qi(Tl teM2 ueM 1 = qn-kQ(n - 1, k - 1) + Q(n - 1, k), which is (19b). Second Proof. Define a map rjJ: 6(M) x 6 a1 x ... x 6ak~ 6n (nO,nl,···,nk)~n by converting the a i i's in n o to the numbers a l + ...

Show that f(n) = [5 - ] [5 - ] ~L n ,p(d)2n / d , dln dodd where ,p denotes Euler's totient function. b. When n is odd, then it can be shown using (a) that f(n) is equal to the number of necklaces (up to cyclic rotation) with n beads, each bead colored black or white. Give a combinatorial proof. ) c. Generalize. For ex am pie, investigate the number of subsets S of 7L/n7L satisfying LiES p(i) == Cl (mod n), where p is a fixed polynomial and Cl E 7L/n7L is fixed. 49 Exercises [2J 28. Let J(n, k) be the number of sequences a l a 2 ...

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A theorem of arithmetic and its proof by Euler L.

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