The Twelyefold Way Gian-Carlo Rota (1932-1999)
Gian-Carlo Rota (1932-1999) The Twelvefold Way
The twelvefold way f:N→M N =n,M=m elements elements of N anyf 1-1 on-to of M distinct distinct identical distinct distinct identical identical identical
The twelvefold way f : N M |N| = n, |M| = m elements of N elements of M any f 1-1 on-to distinct distinct identical distinct distinct identical identical identical
Knuth's version (in TAOCP vol.4A) n balls are put into m bins balls per bin: unrestricted ≤1 ≥1 n distinct balls, m distinct bins mn n identical balls, m distinct bins n distinct balls, m identical bins n identical balls, m identical bins
Knuth’s version (in TAOCP vol.4A) balls per bin: unrestricted ≤ 1 ≥ 1 n distinct balls, m distinct bins n identical balls, m distinct bins n distinct balls, m identical bins n identical balls, m identical bins n balls are put into m bins mn
Tuples {1,2,..,m} [m]=0,1,-1} TTNOE MATCH [mn=m××m I(mj"mr Product rule: finite sets S and T |S×T=|S1·lTI
Tuples mn |[m] n| = Product rule: |S ⇥ T| = |S|·|T| finite sets S and T [m] ⇥ ··· ⇥ [m] ⇤ ⇥ ⌅ n [m] n = [m] = {0, 1,...,m 1} {1, 2,...,m}
Functions count the of functions f:[ml→m [n] [m] (f(1),f(2),.,(n)∈[m]n one-one correspondence [n→[m台[m]m
Functions [n] [m] f : [n] [m] count the # of functions [m] n one-one correspondence [n] [m] ⇥ [m] n (f(1), f(2),...,f(n))