Falling and Rising Factorials m2=(mn=m(m-10m-2).(m-n+)=m-m m! mn=mm=m(m+1)(m+2)…(m+n-1) 4口·1①，43，t夏，里)Q0 Hengfeng Wei (hfweixinju.edu.cn) 2-3 Counting farch12.20207/34
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Falling and Rising Factorials mn = (m)n = m(m − 1)(m − 2)· · ·(m − n + 1) = m! (m − n)! mn¯ = m(n) = m(m + 1)(m + 2)· · ·(m + n − 1) n! = n n = 1 n¯ m n ! = mn n! Hengfeng Wei (hfwei@nju.edu.cn) 2-3 Counting March 12, 2020 7 / 34
Falling and Rising Factorials m2=mn=m(m-1)(m-2)…(m-n+1)=(m-m m! mn=mm=m(m+1)(m+2)…(m+n-1) n!=n2=1n 4口·￥①，43，t夏，3)Q0 Hengfeng Wei (hfweixinju.edu.cn) 2-3 Counting farch12.20207/34
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Falling and Rising Factorials mn = (m)n = m(m − 1)(m − 2)· · ·(m − n + 1) = m! (m − n)! mn¯ = m(n) = m(m + 1)(m + 2)· · ·(m + n − 1) n! = n n = 1n¯ m n ! = mn n! Hengfeng Wei (hfwei@nju.edu.cn) 2-3 Counting March 12, 2020 7 / 34
Falling and Rising Factorials m2=mn=m(m-1)m-2)(m-n+1）=m-m m! m元=mm=m(m+1)(m+2）)…(m+n-1) n!=n2=1n m mz n n! 4口·￥①，43，t夏，3)Q0 Hengfeng Wei (hfweixinju.edu.cn) 2-3 Counting farch12.20207/34
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Falling and Rising Factorials mn = (m)n = m(m − 1)(m − 2)· · ·(m − n + 1) = m! (m − n)! mn¯ = m(n) = m(m + 1)(m + 2)· · ·(m + n − 1) n! = n n = 1n¯ m n ! = mn n! Hengfeng Wei (hfwei@nju.edu.cn) 2-3 Counting March 12, 2020 7 / 34
Iverson Bracket [P]= 1, if P is true; 10, otherwise Kenneth Eugene Iverson (1920~2004) 4口·1①，43，t夏，里)Q0 Hengfeng Wei (hfweignju.edu.cn) 2-3 Counting March12,20208/34
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Hengfeng Wei (hfwei@nju.edu.cn) 2-3 Counting March 12, 2020 8 / 34
Iverson Bracket 1 if P is true; 0 otherwise n≤网= 1, ifn≤m lo. ifn>m Kenneth Eugene Iverson (1920~2004) 4口·1①，43，t夏，里)Q0 Hengfeng Wei (hfweignju.edu.cn) 2-3 Counting March12,20208/34
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Hengfeng Wei (hfwei@nju.edu.cn) 2-3 Counting March 12, 2020 8 / 34