Reference no: EM13664810

Give Obijective arguments to show that each class of objects below
is enumerated by C(n). (All three were selected from the list in Stanley's book.)
1. The number of ways to fully-parenthesize a product of n + 1 factors as if the "multiplication" operation in question were not necessarily associative. For example, there is one way to arenthesize a product of two factors (a1a2), there are two ways to parenthesize a product of three factors ((a1(a2a3)) and ((a1a2)a3)), and there are five ways to parenthesize a product of
four factors:
(a1(a2(a3a4))),(a1((a2a3)a4)),((a1a2)(a3a4)),((a1(a2a3))a4),(((a1a2)a3)a4).
2. Sequences of n 1's and n -1's in which the sum of the first i terms is nonnegative for all i.