Derangement permutationπof[nl i∈[ml,π()卡i "permutations with no fixed point" !n U=Sn symmetric group Ai={ππ(i)=i} ∩A=∑(-1)川14 iEln] ICIn] A虹={πi∈I,π()=i A=(n-I)!
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=SmA:={π|π()=} A虹={π|i∈I,π(i)=}A=(n-I)！ ∩A=∑(-1)川A iEln] IC[n] -)4a-:=-()n- ICIn] n! k！ e
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 of In] derangement:i∈[nl,π(i)≠i generally::π(i)≠ji,π(i2)卡j2,. forbidden positions B[n]x[n] i∈[m],(i,π()主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 permutationπof[m 6 {(i,π()|i∈[n} 4 “A placement of X 1 non-attacking rooks" 9 forbidden positions B[n]x In] derangement: B={(i,)|i∈[nl}
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]}
Chess board For a particular set of forbidden positions Bc[ml×[n 4 No: the of placements of n non-attacking rooks?
Chess board B ⇥ [n] [n] For a particular set of forbidden positions the # of placements of n non-attacking rooks? N0 :