83.47 The Secret Santa problem

92.46 A permutation-ordering algorithm

“…We prove that every non-negative integer N can be written uniquely in the form N = fli.l! + a 2 .2\ + ... + a p .p\ (1) where 0 < a, < /fori = 1, 2, ... ,p. This result depends on the known p summation X i.i\ + 1 = (p + 1)!.…”

“…Clearly the right-hand side of (1) gives a unique non-negative integer. We must now show, conversely, how to write any given non-negative integer in the form (1). This is done by induction on N.…”

“…This in turn will make it easy to pick out those permutations having particular properties (e.g. those which are derangements or those which satisfy the conditions of the Secret Santa problem [1]).…”

Difference equations, determinants and the Secret Santa problem

“…In the light of [2] it now emerges that, by adopting a different approach from that given in [1], we can solve completely every linear second-order difference equation with variable coefficients with reasonable economy of effort. Here, we develop the method of solution for a general second-order equation and illustrate it by applying it to a particularly intractable difference equation, namely that which arises in the secret-Santa problem [3]. {The Secret Santa problem may be stated: All the students in a class write their names on slips of paper which are placed in a box and shuffled.…”

The Secret Santa Problem

“…Some published works in social sciences exist [3]. A scholarly discussion ensued in 1999-2001 in the Mathematical Gazette [7,9,1] focussing on the probability of picking a gift assignment without mutual gifts. This is extended in [8] to deal with more constraints on pairs of people that cannot exchange gifts, and in [13] to include at least a cyclic assignment of given length.…”