Falling and Rising Factorials n严=(n)m=n(m-1)n-2)…(m-m+)= n! (n-m)月 nm=nm)=n(n+1)(n+2)…(m+m-1) 4口·￥①，4三，t更，3)Q0 Jun Ma (majunainju.edu.cn) 2-3 Counting March 9.2022 7/35
Falling and Rising Factorials n m = (n)m = n(n − 1)(n − 2)· · ·(n − m + 1) = n! (n − m)! n m¯ = n (m) = n(n + 1)(n + 2)· · ·(n + m − 1) n! = n n = 1 n¯ n m ! = n! (n − m)!m! = n m m! Jun Ma (majun@nju.edu.cn) 2-3 Counting March 9, 2022 7 / 35
Falling and Rising Factorials n严=(n)m=n(m-l)n-2)…(m-m+)= n! (n-m)月 nm=nm=n(n+1)(n+2)…(m+m-1) n!=n2=1n 4口·￥①，4三，t更，3)Q0 Jun Ma (majunainju.edu.cn) 2-3 Counting March 9.2022 7/35
Falling and Rising Factorials n m = (n)m = n(n − 1)(n − 2)· · ·(n − m + 1) = n! (n − m)! n m¯ = n (m) = n(n + 1)(n + 2)· · ·(n + m − 1) n! = n n = 1n¯ n m ! = n! (n − m)!m! = n m m! Jun Ma (majun@nju.edu.cn) 2-3 Counting March 9, 2022 7 / 35
Falling and Rising Factorials n严=(nm=nm-l)0m-2)…m-m+1)= n! (n-m)月 nm=mm=n(n+1)(n+2)…(n+m-1) n！=n2=1n n! n m (n-m)!m!= m! 4口·￥①，4三，t更，里)Q0 Jun Ma (majunainju.edu.cn) 2-3 Counting March 9.2022 7/35
Falling and Rising Factorials n m = (n)m = n(n − 1)(n − 2)· · ·(n − m + 1) = n! (n − m)! n m¯ = n (m) = n(n + 1)(n + 2)· · ·(n + m − 1) n! = n n = 1n¯ n m ! = n! (n − m)!m! = n m m! Jun Ma (majun@nju.edu.cn) 2-3 Counting March 9, 2022 7 / 35
Iverson Bracket [P] 1, if P is true; 0 otherwise Kenneth Eugene Iverson (1920~2004) 4口·￥①，43，t夏，里Q0 Jun Ma (majunainju.edu.cn) 2-3 Counting March 9.2022 8/35
Iverson Bracket Kenneth Eugene Iverson (1920 ∼ 2004) [P] = ( 1, if P is true; 0, otherwise [n ≤ m] = ( 1, if n ≤ m; 0, if n > m Jun Ma (majun@nju.edu.cn) 2-3 Counting March 9, 2022 8 / 35
Iverson Bracket Pi- 1 if P is true; 10 otherwise n≤m= 1, ifn≤m if n>m Kenneth Eugene Iverson (1920~2004) 4口·￥①，4三，t更，里)Q0 Jun Ma (majunainju.edu.cn) 2-3 Counting March 9.2022 8/35
Iverson Bracket Kenneth Eugene Iverson (1920 ∼ 2004) [P] = ( 1, if P is true; 0, otherwise [n ≤ m] = ( 1, if n ≤ m; 0, if n > m Jun Ma (majun@nju.edu.cn) 2-3 Counting March 9, 2022 8 / 35