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1: 26.13 Permutations: Cycle Notation
See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. A derangement is a permutation with no fixed points. The derangement number, d ( n ) , is the number of elements of 𝔖 n with no fixed points:
26.13.4 d ( n ) = n ! j = 0 n ( 1 ) j 1 j ! = n ! + e 2 e .
2: 26.15 Permutations: Matrix Notation
The number of derangements of n is the number of permutations with forbidden positions B = { ( 1 , 1 ) , ( 2 , 2 ) , , ( n , n ) } . … For the problem of derangements, r j ( B ) = ( n j ) . …
3: Bibliography Z
  • J. Zeng (1992) Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. London Math. Soc. (3) 65 (1), pp. 1–22.