Concrete mathematics : a foundation for computer science

154 BINOMIAL COEFFICIENTS
For example,
0 4 2=-= 4.3 6. 2.1 '
this agrees with our previous enumeration.
We call n the upper index and k the lower index, The indices are
restricted to be nonnegative integers by the combinatorial interpretation, because
sets don’t have negative or fractional numbers of elements. But the
binomial coefficient has many uses besides its combinatorial interpretation,
so we will remove some of the restrictions. It’s most useful, it turns out,
to allow an arbitrary real (or even complex) number to appear in the upper
index, and to allow an arbitrary integer in the lower. Our formal definition
therefore takes the following form:
r(r-l)...(r-kkl) r-k
k(k-l)...(l) = k!’
0,
integer k 3 0;
integer k < 0.
(5.1)
This definition has several noteworthy features. First, the upper index is
called r, not n; the letter r emphasizes the fact that binomial coefficients make
sense when any real number appears in this position. For instance, we have
(,') = (-l)(-2)(-3)/(3.2.1)= -1. There’s no combinatorial interpretation
here, but r = -1 turns out to be an important special case. A noninteger
index like r = -l/2 also turns out to be useful.
Second, we can view (;>I as a kth-degree polynomial in r. We’ll see that
this viewpoint is often helpful.
Third, we haven’t defined binomial coefficients for noninteger lower indices.
A reasonable definition can be given, but actual applications are rare,
so we will defer this generalization to later in the chapter.
Final note: We’ve listed the restrictions ‘integer k 3 0’ and ‘integer
k < 0’ at the right of the definition. Such restrictions will be listed in all
the identities we will study, so that the range of applicability will be clear.
In general the fewer restricti.ons the better, because an unrestricted identity
is most useful; still, any restrictions that apply are an important part of
the identity. When we manipulate binomial coefficients, it’s easier to ignore
difficult-to-remember restrictions temporarily and to check later that nothing
has been violated. But the check needs to be made.
For example, almost every time we encounter (“,) it equals 1, so we can
get lulled into thinking that it’s always 1. But a careful look at definition (5.1)
tells us that (E) is 1 only when n 1: 0 (assuming that n is an integer); when
n < 0 we have (“,) = 0. Traps like this can (and will) make life adventuresome.