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Internet Engineering Czesław Smutnicki Discrete Mathematics – Combinatorics

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

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FUNCTIONS AND DISTRIBUTIONS XY X elements, Y boxes 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

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

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SUBSETS

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

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NUMBER PARTITION

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SECOND TYPE STIRLING NUMBERS

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BELL NUMBERS nBnBn

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

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complexity variance mean receipt number of inversion in -1 o n minus the number of cycles in -1 o n minus the lenght of the maximal increasing subsequence in -1 o measure D A ( , ) D S ( , ) D I ( , ) Move type API NPI INS PERMUTATIONS. DISTANCE MEASURES

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GENERATING PERMUTATIONS. IN ANTYLEX ICOGRAPHICAL 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

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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

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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

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Thank you for your attention DISCRETE MATHEMATICS Czesław Smutnicki

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