Combinatorics Through Guided Discovery, 2004a

3.2. Partitions and Stirling Numbers 61
Problem 149. Explain how to compute the number of functions from a k-
element set K to an n-element set N by using multinomial coefficients. (h)
Problem 150. Explain how to compute the number of functions from a k-
element set K onto an n-element set N by using multinomial coefficients. (h)
• Problem 151. What do multinomial coefficients have to do with expanding
the kth power of a multinomial x 1 + x 2 + ···+ x n ? This result is called the
multinomial theorem. (h)
3.2.3 Stirling Numbers and bases for polynomials
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Problem 152.
(a) Find a way to express n k in terms of k j for appropriate values j. You
may use Stirling numbers if they help you. (h)
(b) Notice that x j makes sense for a numerical variable x (that could range
over the rational numbers, the real numbers, or even the complex
numbers instead of only the nonnegative integers, as we are implicitly
assuming n does), just as x j does. Find a way to express the power x k
in terms of the polynomials x j for appropriate values of j and explain
why your formula is correct. (h)
You showed in Problem 152 how to get each power of x in terms of the falling
factorial powers x j . Therefore every polynomial in x is expressible in terms of a
sum of numerical multiples of falling factorial powers. Using the language of linear
algebra, we say that the ordinary powers of x and the falling factorial powers of x
each form a basis for the “space” of polynomials, and that the numbers S(k, n) are
“change of basis coefficients.” If you are not familiar with linear algebra, a basis
for the space of polynomials3 is a set of polynomials such that each polynomial,
whether in that set or not, can be expressed in one and only one way as a sum of
numerical multiples of polynomials in the set.
◦ Problem 153. Show that every power of x +1is expressible as a sum of
numerical multiples of powers of x. Now show that every power of x (and
thus every polynomial in x) is a sum of numerical multiples (some of which
could be negative) of powers of x +1. This means that the powers of x +1
are a basis for the space of polynomials as well. Describe the change of basis
3The space of polynomials is just another name for the set of all polynomials.