Internet Engineering Czesław Smutnicki Discrete Mathematics – Combinatorics

Internet EngineeringInternet Engineering

Czesław SmutnickiCzesław Smutnicki

Discrete Mathematics Discrete Mathematics – – CombinatorCombinatoricsics

CONTENTS

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

FUNCTIONS AND DISTRIBUTIONS

X Y

X elements, Y boxes

Element can be packed to any box: n-length sequence Each box contains at most one element: set partitionBox contains exactly one element: permutation

mYnXYXf ,,:

K-SUBSETS

Generate all subsets with k-elements from the set of n elements

k

n

123412351236124512461256134513461356145623452346235624563456

ss CCCC ,,...,, 121

1,1,...,1,1,0,0,...,0,0 121121 CCCCCCCC ssss

)1,0,...,1,0,0,1(iC

SUBSETS

0 00001 00012 00103 00114 01005 01016 01107 01118 10009 1001

10 101011 101112 110013 110114 111015 1111

ss CCCC ,,...,, 121

1,1,...,1,1,0,0,...,0,0 121121 CCCCCCCC ssss

)1,0,...,1,0,0,1(iC

SET PARTITION

(1)

(1,2) (1)(2)

(1,2,3) (1,2)(3) (1,3)(2) (1)(2,3) (1)(2)(3)

NUMBER PARTITION

76 15 25 1 14 34 2 14 1 1 13 3 13 2 1 13 1 1 1 12 2 2 12 2 1 1 12 1 1 1 1 11 1 1 1 1 1 1

SECOND TYPE STIRLING NUMBERS

0,0)0,(

0,1),(

0,),1()1,1(),(

nnS

nnnS

nkknkSknSknS

BELL NUMBERS

nkn knSB 0 ),(

n Bn

0 1

1 1

2 2

3 5

4 15

5 52

6 203

7 877

8 4140

PERMUTATIONS. INTRODUCTION

• Permutation• Inverse permutation• Id permutation• Composition• Inversions• Cycles• Sign

4

)1( nn

72

)52)(1( nnn

)( 2nO )(nO )log( nnO

)2(nn HH

nHn

)( 3

1

n

nn 2

complexity

variance

mean

receiptnumber of inversionin -1 o

n minus the numberof cycles in -1 o

n minus the lenght of the maximal increasing subsequence in -1 o

measure DA (, ) DS (, ) DI (, )

Move type API NPI INS

PERMUTATIONS. DISTANCE MEASURES

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; }

elsefor (i=1;i<=m;i++)

{ antylex(m-1); if (i<m) { swap(pi[i],pi[m]); reverse(m-1); }}}

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]); }

}

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;

}

Thank you for your attention

DISCRETE MATHEMATICSCzesław Smutnicki