"Concrete Mathematics"-chaper5-two-term coefficient

The most intuitive combination of the two-item coefficients, which is the number of combinations we commonly use, is to take all possible cases of k elements from n elements, so we naturally get the definition of the following two-term coefficients.

Then we give a more general definition of the two-term coefficients by means of two coefficients with combinatorial meanings:

It is easy to push the identity 3 by the two-polynomial coefficients of the generalized form above.

However, we note that in the derivation of the second identity, we apply the symmetric identity.

The use of symmetric identities requires a non-negative indicator, does this mean that we need to limit the scope of R for the second identity?

The answer is no? The identity is still set on the real number R.

Because first the identity (the accompanying identity of the invariant) is necessarily the non-negative integer r set up, we can find more than the K+1 group of different points, so that the difference between the left and right is 0. And we look at the difference between the left and right in terms of polynomial, they are actually R's K + 1 polynomial. By polynomial theory: a non-zero D-polynomial has at most 0 points different from D, otherwise, it can only indicate that the polynomial is constant 0. So we see that this identity is about the same polynomial as r on the right and left side, so the identity is set for the real number R.

This method of proving is called the polynomial inference method, which is useful in extending the two-term coefficient identities from integers to real numbers.

With the addition identity (Pascal Formula), we can easily simplify the following one and the other, it allows us to solve a regular two-term coefficient of the sum formula, only the calculation of a two-term coefficient, from the point of view of the algorithm time complexity, is O (n) to O (1) optimization.

On the sum of the indicators, here are the combination of explanation: We in the label 0~n N+1 ticket to elect M+1 ticket, this m+1 ticket, the maximum number of k is a C (K, M). where k is the upper indicator of the two-term coefficient.

The connotation of the summation of the indicator is that the Pascal triangle is enlarged upward, and when the index of the two-term coefficient is less than the lower index, the equivalent conversion is usually made by using the identity.

The sum of the products and identities of a series of two-polynomial coefficients is given below.

These identities are obtained through the Vandermonde convolution identity and other identities described above, for their derivation, we will slowly unfold, here for a moment, it is a simple list, in the following instances of simplification, can be taken directly to apply.

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By learning from the above series of basic two-term coefficient identities, let's go on to apply them in combat.

Example 1: The formula of the ratio:

We need to make a simple test of the formula after simplification, which is the basic accomplishment and habit of simplifying operation. Take n = 4, M, 2.

By the formula given in the original formula, the result is 5/3. Using the result of simplification, it is also 5/3.

Example 2:

From the above process we found a very clever approach, how to use the two-parameter equation we discussed earlier to solve the specific problem? --give the parameter specific values in the identity (especially the non-indicative constants in the same type).

For example, we use the identities we just used to give a set of assignments:

"Concrete Mathematics"-chaper5-two-term coefficient