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47-412: The number of derangements of a set of size n is known as the subfactorial of n or the n  th derangement number or n  th de Montmort number (after Pierre Remond de Montmort ). Notations for subfactorials in common use include ! n , D n , d n , or n ¡  . For n > 0 , the subfactorial ! n equals the nearest integer to n !/ e , where n ! denotes

94-450: A floating-point representation with a fixed number of significant digits . In a sequence of calculations, these rounding errors generally accumulate , and in certain ill-conditioned cases they may make the result meaningless. Accurate rounding of transcendental mathematical functions is difficult because the number of extra digits that need to be calculated to resolve whether to round up or down cannot be known in advance. This problem

141-785: A certain sequence of polynomials P n , where P n has degree n . But the above answer for the case r = 2 gives an orthogonality relation, whence the P n 's are the Laguerre polynomials ( up to a sign that is easily decided). In particular, for the classical derangements, one has that ! n = Γ ( n + 1 , − 1 ) e = ∫ 0 ∞ ( x − 1 ) n e − x d x {\displaystyle !n={\frac {\Gamma (n+1,-1)}{e}}=\int _{0}^{\infty }(x-1)^{n}e^{-x}dx} where Γ ( s , x ) {\displaystyle \Gamma (s,x)}

188-491: A derangement is a permutation that leaves none of the n objects fixed, this implies ! n = n ! − | S 1 ∪ ⋯ ∪ S n | = n ! ∑ i = 0 n ( − 1 ) i i !   . {\displaystyle !n=n!-\left|S_{1}\cup \dotsm \cup S_{n}\right|=n!\sum _{i=0}^{n}{\frac {(-1)^{i}}{i!}}~.} On

235-663: A non-recursive formula for the number of derangements of an n -set, as well. For 1 ≤ k ≤ n {\displaystyle 1\leq k\leq n} we define S k {\displaystyle S_{k}} to be the set of permutations of n objects that fix the k  th object. Any intersection of a collection of i of these sets fixes a particular set of i objects and therefore contains ( n − i ) ! {\displaystyle (n-i)!} permutations. There are ( n i ) {\textstyle {n \choose i}} such collections, so

282-479: A tie-breaking rule that is commonly taught and used, namely: If the fractional part of x is exactly 0.5, then y = x + 0.5 if x is positive, and y = x − 0.5 if x is negative. For example, 23.5 gets rounded to 24, and −23.5 gets rounded to −24. This can be more efficient on computers that use sign-magnitude representation for the values to be rounded, because only the first omitted digit needs to be considered to determine if it rounds up or down. This

329-640: A tie-breaking rule that is widely used in many disciplines. That is, half-way values of x are always rounded up. If the fractional part of x is exactly 0.5, then y = x + 0.5 For example, 23.5 gets rounded to 24, and −23.5 gets rounded to −23. Some programming languages (such as Java and Python) use "half up" to refer to round half away from zero rather than round half toward positive infinity . This method only requires checking one digit to determine rounding direction in two's complement and similar representations. One may also round half down (or round half toward negative infinity ) as opposed to

376-504: Is also called convergent rounding , statistician's rounding , Dutch rounding , Gaussian rounding , odd–even rounding , or bankers' rounding . This is the default rounding mode used in IEEE 754 operations for results in binary floating-point formats. By eliminating bias, repeated addition or subtraction of independent numbers, as in a one-dimensional random walk , will give a rounded result with an error that tends to grow in proportion to

423-457: Is any fixed positive integer, and B k {\displaystyle B_{k}} denotes the k {\displaystyle k} -th Bell number . Moreover, the constant implied by the big O -term does not exceed B m + 1 {\displaystyle B_{m+1}} . The problème des rencontres asks how many permutations of a size- n set have exactly k fixed points. Derangements are an example of

470-501: Is given in the table below. There are various other expressions for ! n , equivalent to the formula given above. These include ! n = n ! ∑ i = 0 n ( − 1 ) i i ! {\displaystyle !n=n!\sum _{i=0}^{n}{\frac {(-1)^{i}}{i!}}} for n ≥ 0 {\displaystyle n\geq 0} and where [ x ] {\displaystyle \left[x\right]}

517-430: Is in U and g is in V , and for all a in A , f ( a ) ≠ g ( a ); in other words, where for each f and g , there exists a derangement φ of S such that f ( a ) = φ( g ( a )). Another generalization is the following problem: For instance, for a word made of only two different letters, say n letters A and m letters B, the answer is, of course, 1 or 0 according to whether n = m or not, for

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564-449: Is known as " the table-maker's dilemma ". Rounding has many similarities to the quantization that occurs when physical quantities must be encoded by numbers or digital signals . A wavy equals sign ( ≈ , approximately equal to ) is sometimes used to indicate rounding of exact numbers, e.g. 9.98 ≈ 10. This sign was introduced by Alfred George Greenhill in 1892. Ideal characteristics of rounding methods include: Because it

611-411: Is not usually possible for a method to satisfy all ideal characteristics, many different rounding methods exist. As a general rule, rounding is idempotent ; i.e., once a number has been rounded, rounding it again to the same precision will not change its value. Rounding functions are also monotonic ; i.e., rounding two numbers to the same absolute precision will not exchange their order (but may give

658-433: Is one method used when rounding to significant figures due to its simplicity. This method, also known as commercial rounding , treats positive and negative values symmetrically, and therefore is free of overall positive/negative bias if the original numbers are positive or negative with equal probability. It does, however, still have bias away from zero. It is often used for currency conversions and price roundings (when

705-1546: Is the nearest integer function and ⌊ x ⌋ {\displaystyle \left\lfloor x\right\rfloor } is the floor function . Other related formulas include ! n = ⌊ n ! + 1 e ⌋ ,   n ≥ 1 , {\displaystyle !n=\left\lfloor {\frac {n!+1}{e}}\right\rfloor ,\quad \ n\geq 1,} ! n = ⌊ ( e + e − 1 ) n ! ⌋ − ⌊ e n ! ⌋ , n ≥ 2 , {\displaystyle !n=\left\lfloor \left(e+e^{-1}\right)n!\right\rfloor -\lfloor en!\rfloor ,\quad n\geq 2,} and ! n = n ! − ∑ i = 1 n ( n i ) ⋅ ! ( n − i ) ,   n ≥ 1. {\displaystyle !n=n!-\sum _{i=1}^{n}{n \choose i}\cdot {!(n-i)},\quad \ n\geq 1.} The following recurrence also holds: ! n = { 1 if  n = 0 , n ⋅ ( ! ( n − 1 ) ) + ( − 1 ) n if  n > 0. {\displaystyle !n={\begin{cases}1&{\text{if }}n=0,\\n\cdot \left(!(n-1)\right)+(-1)^{n}&{\text{if }}n>0.\end{cases}}} One may derive

752-682: Is the upper incomplete gamma function . It is NP-complete to determine whether a given permutation group (described by a given set of permutations that generate it) contains any derangements. =1×10 =1×10 =1×10 =2×10 =1×10 =6×10 =2×10 =2.4×10 =9×10 =1.20×10 =4.4×10 =7.20×10 =2.65×10 =5.04×10 ≈1.85×10 ≈4.03×10 ≈1.48×10 ≈3.63×10 ≈1.33×10 ≈3.63×10 ≈1.33×10 ≈3.99×10 Pierre Remond de Montmort Pierre Remond de Montmort

799-404: Is the integer that is closest to 0 (or equivalently, to x ) such that x is between 0 and y (included). For example, 23.2 gets rounded to 24, and −23.2 gets rounded to −24. These six methods are called rounding to the nearest integer . Rounding a number x to the nearest integer requires some tie-breaking rule for those cases when x is exactly half-way between two integers – that is, when

846-413: Is the limit of the probability that a randomly selected permutation of a large number of objects is a derangement. The probability converges to this limit extremely quickly as n increases, which is why ! n is the nearest integer to n !/ e . The above semi-log graph shows that the derangement graph lags the permutation graph by an almost constant value. More information about this calculation and

893-437: Is the odd integer nearest to x . Thus, for example, 23.5 becomes 23, as does 22.5; while −23.5 becomes −23, as does −22.5. This method is also free from positive/negative bias and bias toward/away from zero, provided the numbers to be rounded are neither mostly even nor mostly odd. It also shares the round half to even property of distorting the original distribution, as it increases the probability of odds relative to evens. It

940-502: Is the smallest integer that is not less than x . For example, 23.2 gets rounded to 24, and −23.7 gets rounded to −23. One may also round toward zero (or truncate , or round away from infinity ): y is the integer that is closest to x such that it is between 0 and x (included); i.e. y is the integer part of x , without its fraction digits. For example, 23.7 gets rounded to 23, and −23.7 gets rounded to −23. One may also round away from zero (or round toward infinity ): y

987-745: Is the sum of the counts for the two cases. This gives us the solution to the hat-check problem: Stated algebraically, the number ! n of derangements of an n -element set is ! n = ( n − 1 ) ( ! ( n − 1 ) + ! ( n − 2 ) ) {\displaystyle !n=\left(n-1\right){\bigl (}{!\left(n-1\right)}+{!\left(n-2\right)}{\bigr )}} for n ≥ 2 {\displaystyle n\geq 2} , where ! 0 = 1 {\displaystyle !0=1} and ! 1 = 0. {\displaystyle !1=0.} The number of derangements of small lengths

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1034-404: Is usually better stated as "about 123500 ". On the other hand, rounding of exact numbers will introduce some round-off error in the reported result. Rounding is almost unavoidable when reporting many computations – especially when dividing two numbers in integer or fixed-point arithmetic ; when computing mathematical functions such as square roots , logarithms , and sines ; or when using

1081-489: The factorial of n and e ≈ 2.718281828... is Euler's number . The problem of counting derangements was first considered by Pierre Raymond de Montmort in his Essay d'analyse sur les jeux de hazard . in 1708; he solved it in 1713, as did Nicholas Bernoulli at about the same time. Suppose that a professor gave a test to 4 students – A, B, C, and D – and wants to let them grade each other's tests. Of course, no student should grade their own test. How many ways could

1128-410: The fraction 312/937 with 1/3, or the expression √2 with 1.414 . Rounding is often done to obtain a value that is easier to report and communicate than the original. Rounding can also be important to avoid misleadingly precise reporting of a computed number, measurement, or estimate; for example, a quantity that was computed as 123456 but is known to be accurate only to within a few hundred units

1175-1928: The inclusion–exclusion principle yields | S 1 ∪ ⋯ ∪ S n | = ∑ i | S i | − ∑ i < j | S i ∩ S j | + ∑ i < j < k | S i ∩ S j ∩ S k | + ⋯ + ( − 1 ) n + 1 | S 1 ∩ ⋯ ∩ S n | = ( n 1 ) ( n − 1 ) ! − ( n 2 ) ( n − 2 ) ! + ( n 3 ) ( n − 3 ) ! − ⋯ + ( − 1 ) n + 1 ( n n ) 0 ! = ∑ i = 1 n ( − 1 ) i + 1 ( n i ) ( n − i ) ! = n !   ∑ i = 1 n ( − 1 ) i + 1 i ! , {\displaystyle {\begin{aligned}|S_{1}\cup \dotsm \cup S_{n}|&=\sum _{i}\left|S_{i}\right|-\sum _{i<j}\left|S_{i}\cap S_{j}\right|+\sum _{i<j<k}\left|S_{i}\cap S_{j}\cap S_{k}\right|+\cdots +(-1)^{n+1}\left|S_{1}\cap \dotsm \cap S_{n}\right|\\&={n \choose 1}(n-1)!-{n \choose 2}(n-2)!+{n \choose 3}(n-3)!-\cdots +(-1)^{n+1}{n \choose n}0!\\&=\sum _{i=1}^{n}(-1)^{i+1}{n \choose i}(n-i)!\\&=n!\ \sum _{i=1}^{n}{(-1)^{i+1} \over i!},\end{aligned}}} and since

1222-425: The n  − 1 hats that is not their own. Call the hat which the person P 1 receives h i and consider h i ' s owner: P i receives either P 1 's hat, h 1 , or some other. Accordingly, the problem splits into two possible cases: For each of the n  − 1 hats that P 1 may receive, the number of ways that P 2 , ...,  P n may all receive hats

1269-669: The above limit may be found in the article on the statistics of random permutations . An asymptotic expansion for the number of derangements in terms of Bell numbers is as follows: ! n = n ! e + ∑ k = 1 m ( − 1 ) n + k − 1 B k n k + O ( 1 n m + 1 ) , {\displaystyle !n={\frac {n!}{e}}+\sum _{k=1}^{m}\left(-1\right)^{n+k-1}{\frac {B_{k}}{n^{k}}}+O\left({\frac {1}{n^{m+1}}}\right),} where m {\displaystyle m}

1316-482: The amount is first converted into the smallest significant subdivision of the currency, such as cents of a euro) as it is easy to explain by just considering the first fractional digit, independently of supplementary precision digits or sign of the amount (for strict equivalence between the paying and recipient of the amount). One may also round half to even , a tie-breaking rule without positive/negative bias and without bias toward/away from zero. By this convention, if

1363-566: The conventional round half away from zero . If the fractional part of x is exactly 0.5, then y = x − 0.5 if x is positive, and y = x + 0.5 if x is negative. For example, 23.5 gets rounded to 23, and −23.5 gets rounded to −23. This method treats positive and negative values symmetrically, and therefore is free of overall positive/negative bias if the original numbers are positive or negative with equal probability. It does, however, still have bias toward zero. One may also round half away from zero (or round half toward infinity ),

1410-401: The displacements from the original number x to the rounded value y are all directed toward or away from the same limiting value (0, +∞ , or −∞). Directed rounding is used in interval arithmetic and is often required in financial calculations. If x is positive, round-down is the same as round-toward-zero, and round-up is the same as round-away-from-zero. If x is negative, round-down is

1457-399: The examples below, sgn( x ) refers to the sign function applied to the original number, x . One may round down (or take the floor , or round toward negative infinity ): y is the largest integer that does not exceed x . For example, 23.7 gets rounded to 23, and −23.2 gets rounded to −24. One may also round up (or take the ceiling , or round toward positive infinity ): y

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1504-420: The fraction part of x is exactly 0.5. If it were not for the 0.5 fractional parts, the round-off errors introduced by the round to nearest method would be symmetric: for every fraction that gets rounded down (such as 0.268), there is a complementary fraction (namely, 0.732) that gets rounded up by the same amount. When rounding a large set of fixed-point numbers with uniformly distributed fractional parts,

1551-411: The fractional part of x is 0.5, then y is the even integer nearest to x . Thus, for example, 23.5 becomes 24, as does 24.5; however, −23.5 becomes −24, as does −24.5. This function minimizes the expected error when summing over rounded figures, even when the inputs are mostly positive or mostly negative, provided they are neither mostly even nor mostly odd. This variant of the round-to-nearest method

1598-399: The market appeared to be rising. The problem was caused by the index being recalculated thousands of times daily, and always being truncated (rounded down) to 3 decimal places, in such a way that the rounding errors accumulated. Recalculating the index for the same period using rounding to the nearest thousandth rather than truncation corrected the index value from 524.811 up to 1098.892. For

1645-426: The more common round half up . If the fractional part of x is exactly 0.5, then y = x − 0.5 For example, 23.5 gets rounded to 23, and −23.5 gets rounded to −24. Some programming languages (such as Java and Python) use "half down" to refer to round half toward zero rather than round half toward negative infinity . One may also round half toward zero (or round half away from infinity ) as opposed to

1692-480: The number of ways n letters, each addressed to a different person, can be placed in n pre-addressed envelopes so that no letter appears in the correctly addressed envelope. Counting derangements of a set amounts to the hat-check problem , in which one considers the number of ways in which n hats (call them h 1 through h n ) can be returned to n people ( P 1 through P n ) such that no hat makes it back to its owner. Each person may receive any of

1739-728: The only way to form an anagram without fixed letters is to exchange all the A with B , which is possible if and only if n = m . In the general case, for a word with n 1 letters X 1 , n 2 letters X 2 , ..., n r letters X r , it turns out (after a proper use of the inclusion-exclusion formula) that the answer has the form ∫ 0 ∞ P n 1 ( x ) P n 2 ( x ) ⋯ P n r ( x )   e − x d x , {\displaystyle \int _{0}^{\infty }P_{n_{1}}(x)P_{n_{2}}(x)\cdots P_{n_{r}}(x)\ e^{-x}dx,} for

1786-1376: The other hand, n ! = ∑ i = 0 n ( n i )   ! i {\displaystyle n!=\sum _{i=0}^{n}{\binom {n}{i}}\ !i} since we can choose n - i elements to be in their own place and derange the other i elements in just ! i ways, by definition. From ! n = n ! ∑ i = 0 n ( − 1 ) i i ! {\displaystyle !n=n!\sum _{i=0}^{n}{\frac {(-1)^{i}}{i!}}} and e x = ∑ i = 0 ∞ x i i ! {\displaystyle e^{x}=\sum _{i=0}^{\infty }{x^{i} \over i!}} by substituting x = − 1 {\textstyle x=-1} one immediately obtains that lim n → ∞ ! n n ! = lim n → ∞ ∑ i = 0 n ( − 1 ) i i ! = e − 1 ≈ 0.367879 … . {\displaystyle \lim _{n\to \infty }{!n \over n!}=\lim _{n\to \infty }\sum _{i=0}^{n}{\frac {(-1)^{i}}{i!}}=e^{-1}\approx 0.367879\ldots .} This

1833-415: The professor hand the tests back to the students for grading, such that no student receives their own test back? Out of 24 possible permutations (4!) for handing back the tests, there are only 9 derangements (shown in blue italics above). In every other permutation of this 4-member set, at least one student gets their own test back (shown in bold red). Another version of the problem arises when we ask for

1880-531: The rounding errors by all values, with the omission of those having 0.5 fractional part, would statistically compensate each other. This means that the expected (average) value of the rounded numbers is equal to the expected value of the original numbers when numbers with fractional part 0.5 from the set are removed. In practice, floating-point numbers are typically used, which have even more computational nuances because they are not equally spaced. One may round half up (or round half toward positive infinity ),

1927-540: The same as round-away-from-zero, and round-up is the same as round-toward-zero. In any case, if x is an integer, y is just x . Where many calculations are done in sequence, the choice of rounding method can have a very significant effect on the result. A famous instance involved a new index set up by the Vancouver Stock Exchange in 1982. It was initially set at 1000.000 (three decimal places of accuracy), and after 22 months had fallen to about 520, although

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1974-478: The same value). In the general case of a discrete range, they are piecewise constant functions . Typical rounding problems include: The most basic form of rounding is to replace an arbitrary number by an integer. All the following rounding modes are concrete implementations of an abstract single-argument "round()" procedure. These are true functions (with the exception of those that use randomness). These four methods are called directed rounding to an integer , as

2021-406: The square root of the number of operations rather than linearly. However, this rule distorts the distribution by increasing the probability of evens relative to odds. Typically this is less important than the biases that are eliminated by this method. One may also round half to odd , a similar tie-breaking rule to round half to even. In this approach, if the fractional part of x is 0.5, then y

2068-435: The wider field of constrained permutations. For example, the ménage problem asks if n opposite-sex couples are seated man-woman-man-woman-... around a table, how many ways can they be seated so that nobody is seated next to his or her partner? More formally, given sets A and S , and some sets U and V of surjections A → S , we often wish to know the number of pairs of functions ( f ,  g ) such that f

2115-579: Was a French mathematician . He was born in Paris on 27 October 1678 and died there on 7 October 1719. His name was originally just Pierre Remond. His father pressured him to study law, but he rebelled and travelled to England and Germany, returning to France in 1699 when, upon receiving a large inheritance from his father, he bought an estate and took the name de Montmort. He was friendly with several other notable mathematicians, and especially Nicholas Bernoulli , who collaborated with him while visiting his estate. He

2162-627: Was elected a fellow of the Royal Society in 1715, while traveling again to England, and became a member of the French Academy of Sciences in 1716. De Montmort is known for his book on probability and games of chance, Essay d'analyse sur les jeux de hazard , which was also the first to introduce the combinatorial study of derangements . He is also known for naming Pascal's triangle after Blaise Pascal , calling it "Table de M. Pascal pour les combinaisons." Another of de Montmort's interests

2209-481: Was the subject of finite differences . He determined in 1713 the sum of n terms of a finite series of the form where Δ is the forward difference operator , a theorem which seems to have been independently rediscovered by Goldbach in 1718. Nearest integer function Rounding or rounding off means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing $ 23.4476 with $ 23.45 ,

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