Combinatorics Through Guided Discovery, 2004a

179
identical ones. If we permute the distinct books before replacement, does
that affect the final outcome? There are other ways to solve this problem.
130. Do you see a relationship between compositions and something else we have
counted already?
131. If we line up k identical books, how many adjacencies are there in between
books?
133. Imagine taking a stack of k books, and breaking it up into stacks to put into
the boxes in the same order they were originally stacked. If you are going
to use n boxes, in how many places will you have to break the stack up into
smaller stacks, and how many ways can you do this?
Additional Hint: How many different bookcase arrangements correspond to
the same way of stacking k books into n boxes so that each box has at least
one book?
134. The number of partitions of [k] into n parts in which k is not in a block relates
to the number of partitions of k − 1 into some number of blocks in a way that
involves n. With this in mind, review how you proved Pascal’s (recurrence)
equation.
137. What if the question asked about six sandwiches and two distinct bags? How
does having identical bags change the answer?
138. What are the possible sizes of parts?
139. Suppose we make a list of the k items. We take the first k 1 elements to be the
blocks of size 1. How many elements do we need to take to get k 2 blocks of
size two? Which elements does it make sense to choose for this purpose?
141. To see how many broken permutations of a k element set into n parts do
not have k is a part by itself, ask yourself how many broken permutations
of [7] result from adding 7 to the one of the two permutations in the broken
permutation {14, 2356}.
142.b. Here it is helpful to think about what happens if you delete the entire block
containing k rather than thinking about whether k is in a block by itself or
not.
143. You can think of a function as assigning values to the blocks of its partition.
If you permute the values assigned to the blocks, do you always change the
function?
144. The Prüfer code of a labeled tree is a sequence of n − 2 entries that must
be chose from the vertices that do not have degree 1. The sequence can be