derangement

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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 $$d(n)=n!\sum _{j=0}^n{(-1)}^j\frac{1}{j!}=\lfloor \frac{n!+\mathrm{e}-2}{\mathrm{e}}\rfloor .$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $j$: nonnegative integer, $n$: nonnegative integer and $d(n)$: derangement number

Permalink:

http://dlmf.nist.gov/26.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.13 and Ch.26

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2: 26.15 Permutations: Matrix Notation

… ►The number of derangements of $n$ is the number of permutations with forbidden positions $B=\{(1,1),(2,2),\mathrm{\dots },(n,n)\}$. … ►For the problem of derangements, $r_j(B)=\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)$. …

3: Bibliography Z

… ►

Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. London Math. Soc. (3) 65 (1), pp. 1–22.

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External Links:

ISSN 0024-6115, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

Encodings:

BibTeX

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