X subject to certain restrictions. There will be three restrictions on the functions themselves and four restrictions on when we consider two functions to be the same. This gives a total of twelve counting problems, and their solution is called the TlVelvejold Way .

Thus [1 - ] denotes an utterly trivial problem, and [5 -] denotes an unsolved problem that has received little attention and may not be too difficult. A rating of [2 + ] denotes about the hardest problem that could be reasonably assigned to a class of graduate students. A few students may be capable of solving a [3 - ] problem, while almost none could solve a [3] in a reasonable period of time. Of course the ratings are subjective, and there is always the possibility of an overlooked simple proof that would lower the rating.

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 S(n, k) denote a Stirling number of the second kind. The generating function S(n, k)x = x k /(l - x)(1 - 2x)'" (1 - kx) implies the identity n Ln S(n, k) = L I al-12a2-1 ... k ak - 1 , (36) the sum being over all compositions a 1 + ... + a k = n. 4. That is, we want to associate with each partition 1t of [n] into k blocks a composition a 1 + ...