Random Necklaces Require Fewer Cuts

SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1381-1408, June 2024.
Abstract. It is known that any open necklace with beads of [math] types, in which the number of beads of each type is divisible by [math], can be partitioned by at most [math] cuts into intervals that can be distributed into [math] collections, each containing the same number of beads of each type. This is tight for all values of [math] and [math]. Here, we consider the case of random necklaces, where the number of beads of each type is [math]. Then the minimum number of cuts required for a “fair” partition with the above property is a random variable [math]. We prove that for fixed [math] and large [math], this random variable is at least [math] with high probability. For [math], fixed [math], and large [math], we determine the asymptotic behavior of the probability that [math] for all values of [math]. We show that this probability is polynomially small when [math], is bounded away from zero when [math], and decays like [math] when [math]. We also show that for large [math], [math] is at most [math] with high probability and that for large [math] and large ratio [math], [math] is [math] with high probability.