Derangement permutation r of [n] i∈[m,π(2)≠i "permutations with no fixed point" n U permutations of [n
Derangement ⇤i [n], (i) ⇥= i permutation of [n] “permutations with no fixed point” U : permutations of [n] !n
Derangement permutation r of [n] i∈[m,π()卡i "permutations with no fixed point" !n U=Sn symmetric group A,={π|π(i)=i} ∩A=∑(-1)川|A IEIn] Ar={πi∈I,π()=} Ar=(m-I)月
U = Sn Derangement ⇤i [n], (i) ⇥= i permutation of [n] “permutations with no fixed point” Ai = { | (i) = i} AI = { | ⇥i I, (i) = i} |AI | = (n |I|)! I[n] (1)|I| |AI | i[n] Ai = symmetric group !n
Derangement U=Sn A;={π|π()=} A虹={π|i∈I,π(2)=}Ar=(n-|I)！ ∩A-(-1)川A iEln] ICIn] = (-1)川(m-I0！=】 -（份）- IC[n] n. k！ ≈ o =0
Derangement Ai = { | (i) = i} AI = { | ⇥i I, (i) = i} |AI | = (n |I|)! = I[n] (1)|I| (n |I|)! = ⇤ n k=0 (1)k n k ⇥ (n k)! = n! n k=0 (1)k k! n! e I[n] (1)|I| |AI | i[n] Ai = U = Sn
Permutations with restricted positions permutation r of [n derangement: i∈[ml,π()卡i generally:π(i1)≠j1,π(i2)≠j2,. forbidden positions Bc[m×[ml i∈[nl,(i,π(i)年B
Permutations with restricted positions derangement: permutation of [n] ⇤i [n], (i) ⇥= i generally: (i1) = j1, (i2) = j2,... B ⇥ [n] [n] ⇤i [n], (i, (i)) ⇥ B forbidden positions
Chess board a b c d e f g h 8 × 8 permutationπof[m × 7 6 X 宣 6 {(i,π()|i∈[n 5 X 5 4 X 4 3 × X “A placement of 宣 1 non-attacking rooks'” b d e g h forbidden positions B[n]x In] derangement: B={(i,)|i∈[m}
Chess board permutation of [n] {(i, (i)) | i [n]} “A placement of non-attacking rooks” forbidden positions B ⇥ [n] [n] derangement: B = {(i, i) | i [n]}