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The subfactorial or left factorial, written \(!n\), is the number of ways that n objects can be arranged where no object appears in its natural position (known as "derangements.") There are many formulas for \(!n\): \begin{eqnarray*} !n &=& n! \displaystyle\sum^{n}_{i = 0} \frac{(-1)^i}{i!}\\ &=& \displaystyle\sum^{n}_{i = 0} i! (-1)^{n - i} \binom{n}{i}\\ &=& \displaystyle\frac{\Gamma(n + 1, -1)}{e}\\ &=& \left[\frac{n!}{e}ight] ext{ (only for $n > 0$)} \end{eqnarray*} In the last formula, [n] means the nearest integer to n. (It is a direct consequence of the first formula — the summation converges to \(1/e\).)

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  • Subfactorial
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  • The subfactorial or left factorial, written \(!n\), is the number of ways that n objects can be arranged where no object appears in its natural position (known as "derangements.") There are many formulas for \(!n\): \begin{eqnarray*} !n &=& n! \displaystyle\sum^{n}_{i = 0} \frac{(-1)^i}{i!}\\ &=& \displaystyle\sum^{n}_{i = 0} i! (-1)^{n - i} \binom{n}{i}\\ &=& \displaystyle\frac{\Gamma(n + 1, -1)}{e}\\ &=& \left[\frac{n!}{e}ight] ext{ (only for $n > 0$)} \end{eqnarray*} In the last formula, [n] means the nearest integer to n. (It is a direct consequence of the first formula — the summation converges to \(1/e\).)
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abstract
  • The subfactorial or left factorial, written \(!n\), is the number of ways that n objects can be arranged where no object appears in its natural position (known as "derangements.") There are many formulas for \(!n\): \begin{eqnarray*} !n &=& n! \displaystyle\sum^{n}_{i = 0} \frac{(-1)^i}{i!}\\ &=& \displaystyle\sum^{n}_{i = 0} i! (-1)^{n - i} \binom{n}{i}\\ &=& \displaystyle\frac{\Gamma(n + 1, -1)}{e}\\ &=& \left[\frac{n!}{e}ight] ext{ (only for $n > 0$)} \end{eqnarray*} In the last formula, [n] means the nearest integer to n. (It is a direct consequence of the first formula — the summation converges to \(1/e\).) The first few values of !n for n = 0, 1, 2, 3, 4, 5, etc. are 1, 0, 1, 2, 9, 44, 265, 1854, 14833, ... In base 10, only one number is equal to the sum of the subfactorials of its digits: 148349 = !1 + !4 + !8 + !3 + !4 + !9.
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