1. Opracowanie językowe dr inż. J. Jarnicki
Internet Engineering
Czesław Smutnicki
Discrete Mathematics – Combinatorics

2. CONTENTS Functions and distributions Combinatorial objects K-subsets
Sequences
Set partitions
Number partitions
Stirling numbers
Bell numbers
Permutations
Set on/off rule

3. FUNCTIONS AND DISTRIBUTIONS
X elements, Y boxes X Y
Element can be packed to any box: n-length sequence
Each box contains at most one element: set partition
Box contains exactly one element: permutation

4. K-SUBSETS
Generate all subsets with k-elements from the set of n elements
1234
1235
1236
1245
1246
1256
1345
1346
1356
1456
2345
2346
2356
2456
3456

5. SUBSETS
0 0000
1 0001
2 0010
3 0011
4 0100
5 0101
6 0110
7 0111
8 1000
9 1001

6. SET PARTITION (1) (1)(2) (1,2) (1,3)(2) (1)(2,3) (1)(2)(3) (1,2,3)
(1,2)(3)

7. NUMBER PARTITION 7
6 1
5 2
5 1 1
4 3
4 2 1
3 3 1

8. SECOND TYPE STIRLING NUMBERS

9. BELL NUMBERS n Bn 1 2 3 5 4 15 52 6
203 7
877 8
4140

10. PERMUTATIONS. INTRODUCTION
Inverse permutation
Id permutation
Composition
Inversions
Cycles
Sign

11. PERMUTATIONS. DISTANCE MEASURES
Move type API NPI INS
DA (, )
DS (, )
DI (, )
measure
n minus the number of cycles in -1 o 
n minus the lenght of the maximal increasing
subsequence in -1 o 
number of inversion
in -1 o 
receipt
mean
variance
complexity

12. GENERATING PERMUTATIONS. IN ANTYLEXICOGRAPHICAL ORDER
void swap(int& a, int& b) { int c=a; a=b; b=c; }
void reverse(int m) { int i=1,j=m; while (i<j) swap(pi[i++],pi[j--]); }
void antylex(int m)
{ int i;
if (m==1)
{ for (i=1;i<=m;i++) cout << pi[i] << ' '; cout << endl; }
else
for (i=1;i<=m;i++)
{ antylex(m-1); if (i<m) { swap(pi[i],pi[m]); reverse(m-1); }} }

13. GENERATING PERMUTATIONS. MINIMAL NUMBER OF TRANSPOSITIONS
void swap(int& a, int& b) { int c=a; a=b; b=c; }
int B(int m, int i) { return (!(m%2)&&(m>2))?(i<(m-1)?i:m-2):m-1; }
void perm(int m)
{ int i;
if (m==1) { for (i=1;i<=n;i++) cout << pi[i] << ' '; cout << endl; }
else for (i=1;i<=m;i++) { perm(m-1); if (i<m) swap(pi[B(m,i)],pi[m]); } }

14. GENERATING PERMUTATIONS. MINIMAL NUMBER OF ADJACENT SWAPS
void swap(int& a, int& b) { int c=a; a=b; b=c; }
void permtp(int m)
{ int i,j,x,k;
int *c=new int[m+1],*pr=new int[m+1];
for (i=1;i<m;i++) { pi[i]=i; c[i]=1; pr[i]=1; }
c[m]=0; for (j=1;j<=n;j++) cout << pi[j] << ' '; cout << endl;
i=1;
while (i<m)
{ i=1; x=0;
while (c[i]==(m-i+1)) { pr[i]=!pr[i]; c[i]=1; if (pr[i]) x++; i++; }
if (i<m)
{ k=pr[i]?c[i]+x:m-i+1-c[i]+x; swap(pi[k],pi[k+1]); c[i]++;
for (j=1;j<=n;j++) cout << pi[j] << ' '; cout << endl; }
delete[] c; delete[] pr;

15. Thank you for your attention
DISCRETE MATHEMATICS
Czesław Smutnicki