5.1 Pascal triangle
In other words, Pascal's Triangle.
Its properties can be found in 3:
1) \ (\ {n} stage {k} = \ n {stage} {nk} \)
2） \(\sum\limits_{k=0}^n \binom{n}{k}=2^n\)
3) Pascal's Triangle term \ (\ binom {n} { k} \) values represent the best from the point to which a number of paths.
5.2 binomial theorem
Binomial theorem
Set \ (n-\) is a positive integer, for all \ (X \) and \ (Y \) have \ ((x + y) ^ n = \ sum \ limits_ {k = 0} ^ n \ binom {n} {k} x ^ ky ^ { nk} \)
In \ (y = 1 \) special circumstances when: \ ((. 1 + X) ^ n-= \ SUM \ limits_ {K = 0} ^ n-\ Binom {n-} {K} X ^ K \) , also commonly used formula.
On common identity binomial coefficients:
1) \ (k \ stage {n} to {n} = \ {n-stage 1} {k-1} \)
The equation can be opened with a card defined.
2） \(\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n}=2^n\)
Order \ (. 1 X =, Y =. 1 \) , can be substituted into the binomial theorem Syndrome. (May also be combined inference)
3) 交错和\ (\ {n} stage 0} {- \ {n} pairing of {1} \ {n} stage {2} - \ stage {n} {3} + ... + (- 1) ^ n \ n {stage} {n} = 0 \)
Can also be written \ (\ binom {n} { 0} + \ binom {n} {2} + ... = \ binom {n} {1} + \ binom {n} {3} + ... = 2 ^ {n-1} \)
Order \ (X =. 1, Y -1 = \) , can be substituted into the binomial theorem Syndrome. (May also be combined inference)
4） \(1\binom{n}{1}+2\binom{n}{2}+...+n\binom{n}{0}=n2^{n-1}\)
Using \ (K \ Binom {n-} {K} = n-\ Binom {n--. 1} {K-. 1} \) , the left can be written as \ (n \ binom {n- 1} {0} + n \ binom . 1-n-{} {}. 1 + ... + n-\. 1-n-Binom {} {} =. 1-n-N2-n-^ {}. 1 \) .
5) using a continuous derivation and on (X \) \ obtained by multiplying \ (\ sum \ limits_ {k = 1} ^ nk ^ p \ binom {n} {k} \) on a positive integer \ (P \) of identity
由 \((1+x)^n=\sum\limits_{k=0}^n \binom{n}{k} x^k\)
Both sides of the \ (X \) derivative: \ (n-(. 1 + X) ^ {n--. 1} = \ SUM \ limits_ {K = 0} ^ n-\ Binom {n-} {K} KX ^ {K-. 1 } \)
(Order \ (x = 1 \) can be obtained: \ (N2 ^ {n--. 1} = \ SUM \ limits_ {K = 0} ^ NK \ Binom {n-} {K} = \ SUM \ limits_ {K =. 1 NK ^} \ {n-Binom {K}} \) )
Both sides of the passenger \ (X \) to give: \ (NX (. 1 + X). 1-n-^} = {\ SUM \ limits_ {0} ^ n-K = \ {n-Binom KX} {K} ^ K \)
Both sides of the \ (X \) derivative: \ (n-((. 1 + X) ^ {n--. 1} + X (n--. 1) (. 1 + X) ^ {n--2}) = \ SUM \ limits_ { k = 0} ^ n \ binom {n} {k} k ^ 2x ^ {k-1} \)
(Order \ (x = 1 \) can be obtained: \ (n-(n-+. 1) 2 ^ {n--2} = \ SUM \ limits_ {K = 0} ^ NK ^ 2 \ Binom {n-} {K} = \ SUM \ limits_ K = {}. 1 NK ^ 2 ^ \ n-Binom {} {} K \) )
6) Vandermonde convolution equation \ (\ sum \ limits_ {k = 0} ^ n \ binom {m1} {k} \ binom {m2} {nk} = \ binom {m1 + m2} {n} \)
Special form \ (\ sum \ limits_ {k = 0} ^ n \ binom {n} {k} ^ 2 = \ binom {2n} {n} \)
Using a combination of reasoning Proof:
set \ (S \) to have \ (m2 \ m1 +) set of elements, the \ (\ binom {m1 + m2 } {n} \) count is \ (S \) a \ (n-\) the number of sets subspace.
The \ (S \) is divided into \ (A, B \) two subsets, where \ (| A | = M1, | B | = M2 \) .
Consider each \ (S \) a \ (n-\) Yuanzi set comprising \ (K \) a \ (A \) elements and \ (NK \) a \ (B \) elements, \ (K \ ) of \ (0 \) to the \ (n-\) integer.
The \ (S \) a \ (n-\) subspace may be set according to (K \) \ magnitude is divided into\ (n + 1 \) portions, and the size of each portion is \ (\ binom {m1} {
k} \ binom {m2} {nk} \) by the adder principle available, \ (\ SUM \ limits_ {K = 0} ^ n \ binom { m1} {k} \ binom {m2} {nk} = \ binom {m1 + m2} {n} \)
Generalized binomial coefficients
\(\binom{r}{k}\) ，\(r\in R,k\in Z\)
\[ \begin{equation*} \binom{r}{k}= \begin{cases} \frac{r(r-1)...(r-k+1)}{k!}& k \leq 1\\ 1& k=0\\ 0& k\leq -1 \end{cases} \end{equation*} \]
Formula \ (\ binom {r} { k} = \ binom {r-1} {k} + \ binom {r-1} {k-1} \) and \ (k \ binom {r} {k} = r \ binom {r-1} {k-1} \) still holds.
Pascal recursion formula obtained by the two summation formula:
1） \(\binom{r}{0}+\binom{r+1}{1}+..\binom{r+k}{k}=\binom{r+k+1}{k}\)
The first formula applied to the left \ (\ binom {r} { - 1} \) to permit.
2) \ (\ {0} stage {k} \ {1} stage {k} + .. \ stage {n} {k} = \ {stage n + 1} {k + 1} \)
The first formula applied to the left \ (\ binom {0} { k + 1} \) to permit.
Single peak of 5.3 binomial coefficients
Binomial coefficient sequence \ (\ binom {n} { 0}, \ binom {n} {1}, ..., \ binom {n} {n} \) is unimodal sequence of greatest \ (\ binom {n} {\ lfloor n / 2 \ rfloor} = \ binom {n} {\ lceil n / 2 \ rceil} \)
Polynomial Theorem 5.4
Symbol hit too hard, a little ......
5.5 Newton's binomial theorem
Several export type is very important in the generation function.
5.6 More on posets
Theorem 5.6.1 ( \ (Dilworth \) theorem "duality" theorem)
Provided ( \ (X-, \ Leq \) ) is a finite set of partial order, provided \ (R & lt \) is the maximum size of the chain. The \ (X-\) may be divided into \ (R & lt \) Article anti-chain, not divided into less than \ (R & lt \) Article reverse strand.
\ (Dilworth \) Theorem
Provided ( \ (X-, \ Leq \) ) is a finite set of partial order, provided \ (m \) is the maximum size of the reverse strand. The \ (X-\) may be divided into \ (m \) strand, divided into not less than \ (m \) chains.