Appendix
Permutations and Combinations
PC -1
Permutations


# of m-permutations of n items:
n!
n
Pm 
n  m !
3-permutations of {1,2,3,4} in lexicographic
order:
1 2 3, 1 2 4, 1 3 2, 1 3 4, 1 4 2, 1 4 3,
2 1 3, 2 1 4, 2 3 1, 2 3 4, 2 4 1, 2 4 3,
3 1 2, 3 1 4, 3 2 1, 3 2 4, 3 4 1, 3 4 2,
4 1 2, 4 1 3,4 2 1, 4 2 3, 4 3 1, 4 3 2.
PC -2
Generating permutations
lexicographically

m-permutations of {1,2,...,n}:
first permutation: 1 2 ... m
last permutation: n n-1 ... n-m+1


All permutations except the last permutation are
updatable ( can be used to generate the next
permutation).
The updatable condition for a permutation (P1P2 ...
Pm):
 j, Pi < j  n and j is not used at the left of Pi.
PC -3
The updating algorithm

Let Pi be the rightmost element and j be the
smallest index satisfying the updatable condition.
Pi  j
Pi+1  the first position which is not used.


Pi+k  the kth position which is not used.


Pm  the (m-i)th position which is not used.
PC -4
Example of updating

5-permutations of {1,2,3,4,5,6,7,8,9}:


81362
81364
81365
81367
81369
81372

81379
81392

81397
81423

Pi=2,
Pi=4,
Pi=5,
Pi=7,
Pi=6,
j=4
j=5
j=7
j=9
j=7
Pi=7, j=9
Pi=3, j=4
PC -5
Ranking permutations



Let N={1,2,3,…,n}
The rank sequence {r1, r2, ..., rm} of permutation
(P1P2...Pm):
ri is rank of Pi in N-{P1 , P2 ,... , Pi-1},
where the smallest rank is 0.
r1r2...rm can be seen as a mixed radix integer:
0  rm  n-m
0  rm-1  n-m+1
(n-m+1 digits)
(n-m+2 digits)
:
0  r2  n-2
0  r1  n-1

rankp(P1,P2,...,Pm) = [
(n-1 digits)
m1
mi 1
 r  n  i  j ]  r
i 1
i
j 0
m
PC -6
Example of ranking

3-permutations of {1,2,3,4}
P1P2P3
--------123
124
132
134
142
143
213
214
231
234
241
243
:
r1 r 2 r3
------000
001
010
011
020
021
100
101
110
111
120
121
rankp
-----------------------0
1
2
3
4
5
6
7
8
9
10
11=1×3×2+2×2+1
PC -7
Unranking permutations

Given d = rankp( P1, P2, ..., Pm ), how to
obtain ( P1 P2 ... Pm) from d?
m  j 1
m  j 1
i 1


ri  (d   rj  n  j  k ) /  n  j  k 
j 1
k 0
k 0


for i =1, 2, ..., m

d=11 in 3-permutations of {1,2,3,4}
2
11
3
5
1
1
r3
2
r2
r1
PC -8
Unranking permutations

Pi = ri + i - di where di is the smallest nonnegative
integer such that
i 1
di   ri  i  di  Pj , Pi  Pj , j  i
j 1

d=11
P1= r1 + 1 - d1 = 1+1-0 =2
P2= r2 + 2 - d2 = 2+2-d2 = 4
P3= r3 + 3 - d3 = 1+3-d3 = 1+3-1=3
PC -9
Combinations

# of m-combinations of n items:
n

n!
Cm 
n  m ! m!
3-combinations of {1,2,3,4} in
lexicographic order:
1 2 3, 1 2 4, 1 3 4, 2 3 4
PC -10
Generating combinations
lexicographically

m-combinations of {1, 2, ..., n}:

first combination:12...m

last combination:(n-m+1)(n-m+2)...n

The updatable condition for a combination
(c1c2...cm):
 j, 1 j  m, cj < n - m + j
PC -11
The updating algorithm




step1: Find the largest j satisfying the updatable
condition.
step2: cj  cj + 1
step3: cj+1  cj + 1, cj+2  cj+1 + 1, ...,
cm  cm-1 + 1
Example: 3-combinations of {1, 2, 3, 4, 5, 6}

1
1
1
1
3
3
3
4
4
5
6
5
PC -12
Ranking combinations

3-combinations of {1, 2, 3, 4, 5, 6}
c1 c2 c3
-------1 2 3
1 2 4
1 2 5
1 2 6
1 3 4
1 3 5
1 3 6
1 4 5
1 4 6
1 5 6
rankc
--------0
1
2
3
4
5
6
7
8
9
c1 c2 c3
-------2 3 4
2 3 5
2 3 6
2 4 5
2 4 6
2 5 6
3 4 5
3 4 6
3 5 6
4 5 6
rankc
--------10
11
12
13
14
15
16
17
18
19
PC -13
Ranking rules


3-combinations of {1, 2, 3, 4, 5, 6}
# of combinations greater than (2 3 4):




Fix nothing, 3-combinations of {3,4,5,6}=4
Fix (2), 2-combinations of {4,5,6}=3
Fix (2 3), 1-combinations of {5,6}=2
Rankc(2 3 4)=19-(4+3+2)=10
PC -14
Unranking combinations


Given g = rankc( c1, c2, ..., cm ), how to obtain ( c1
c2 ... cm) from g?
g=10 in 3-combinations of {1, 2, 3, 4, 5, 6}


(C36  1)  10  19  10  9
C35  10  9  C34  4 - - - - -  (2)
945
5  C23  3 - - - --  (2 3)
53 2
C12  2 - - - --  (2 3 4)
PC -15