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A102262 Numerators of probabilities in gift exchange problem with n people. 1
0, 1, 5, 19, 203, 4343, 63853, 58129, 160127, 8885501, 1500518539, 404156337271, 16040576541971, 1694200740145637, 24047240650458731, 22823917472900053, 2511014355032164231, 143734030512459889193, 49611557898193759558813 (list; graph; refs; listen; history; text; internal format)



This is a version of the Secret Santa game.

n friends organize a gift exchange. The n names are put into a hat and the first person draws one. If she picks her own name, then she returns it to the bag and draws again, repeating until she has a name that is not her own. Then the second person draws, again returning his own name if it is drawn. This continues down the line. What is the probability p(n) that when the n-th person draws, only her own name will be left in the bag?

I heard about the problem from Gary Thompson at Grove City College in PA.


Table of n, a(n) for n=2..20.

Math Forum at Drexel, A variant on the "Secret Santa"


p(n) = Sum_{i=1..n-2} t(n,i)/(n-1)!^2,

where t(n,i) = (n-2)*i^2/(i-1)*t(n-1,i-1)-(n-i-2)*t(n-1,i) for 1<i<n-1;

t(n,1) = (-1)^(n-1)*(n-1)!/2 for i=1 and n>2;

t(n,i) = 0 otherwise. - Jon E. Schoenfield, Sep 30 2006

As n increases, p(n) approaches 1/(n + log(n) + EulerGamma), where EulerGamma = 0.5772156649015... (the Euler-Mascheroni constant). - Jon E. Schoenfield, Sep 30 2006


p(2) through p(10) are 0, 1/4, 5/36, 19/144, 203/1800, 4343/43200, 63853/705600, 58129/705600, 160127/2116800.


Cf. A102263.

Sequence in context: A145935 A024529 A106991 * A123281 A135171 A058765

Adjacent sequences:  A102259 A102260 A102261 * A102263 A102264 A102265




Jerrold Grossman, Feb 17 2005


More terms from Jon E. Schoenfield, Sep 30 2006



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Last modified July 30 05:44 EDT 2021. Contains 346349 sequences. (Running on oeis4.)