The Distribution of the Size of the Intersection of a k-Tuple of Intervals

Let (I1; I2; :::; Ik) be a random k-tuple of subintervals of the discrete interval [1; n], and Ln the random variable that measures the size of their insersection. We derive the exact and asymptotic distribution of Ln under the assumption of equally likely drawn k-tuples. The enumeration of such k-tuples and re…nements of the given statistic lead to interesting relations to other topics, like octahedral numbers and bipartite graphs. 1 Introduction How do we decide whether a sequence of subintervals from [1; n] = f1; : : : ; ng is “random”, i.e., independently and equally likely drawn from all n+1 2 k subintervals, if all we get to see is the size of their intersection? This paper was motivated by investigating the distribution of the intersection size of k subintervals of [1; n]. For example, if 5 subintervals from [1; 10] intersect in 4 or more points, we can be 99:5% certain that they are not randomly generated. Alternatively, suppose we can draw the subintervals one after the other. If n is larger than 4, and the …rst 10 intervals still have a nonempty intersection, we can be more than 99% sure that the intervals are not random. We will see in Section 6 that the probability of drawing k intersecting intervals approaches 2= 2k k 2 k p k for n!1. Let n;l := f(I1; I2; :::; Ik) j Ij is a subinterval of [1; n] for all j 2 [1; k], and Tkj=1 Ij = lg. The enumeration begins with the basic observation in Section 2 that n;l = k n+1 l;1 for l > 0. Whenever feasible, we will therefore write j instead of k j;1. In this notation Pr (The size of the intersection of 5 subintervals of [1; 10] is 4 or larger) = 11 2 5 10 X l=4 11 l;1 = 0:00474 using one of the many formulas we will derive for n;1 . A Short Table of n k # n = 1 2 3 4 5 6 7 2 1 6 19 44 85 146 231 3 1 14 87 340 1001 2442 5215 4 1 30 355 2300 10 213 35 162 100 935 5 1 62 1383 14644 97145 469 146 1803 007 In Section 6.1 we determine the expectation of Ln, well approximated by 2 (n+ 1) k!k!= n (2k + 1)! , and the variance. In terms of subsets instead of subintervals a related problem is discussed in Stanley’s Enumerative Combinatorics I [5, Example 1.1.16]; the additional structure gained from intervals makes n a very interesting object to study, with surprisingly many aspects and re…nements. Experimenting with small values of k was fruitful; we found by ad hoc arguments and algebra that j nj = n 1 + 4 n 2 + 4 n 3 j nj = n 1 + 12 n 2 + 48 n 3 + 72 n 4 + 36 n 5 j nj = n 1 + 28 n 2 + 268 n 3 + 1056 n 4 + 1968 n 5 + 1728 n 6 + 576 n 7 A combinatorial interpretation of the coe¢ cients (called c (p; k)) of n p in the expansion of n will be given in Section 4.1. We found the (octahedral) numbers j nj especially noteworthy; they are discussed in Section 3. Expanding n in powers of n shows another interesting feature: j nj = 1 3n+ 2 3 n, j nj = 1 5n+ 1 2 n + 3 10 n, j nj = 4 35n 7 + 4 15 n + 2 5 n + 23 105 n, j nj = 37n 5 126 n + 1 3 n + 5 21 n + 5 126 n Only odd powers of n occur in those expansion; however, the negative coe¢ cient in j nj discourages a search for combinatorial signi…cance. It turns out that the Bernoulli numbers Bk are to blame (Section 5); on the other hand, they give us a rather precise approximation of our numbers, n ((n+1) 2n2k+1+(n 1))k!k! (2k+1)! 2((n+1) 2nk+(n 1))Bk+1 k+1 for odd k, and a similar approximation for even k (Corollary 9).The numerical experiments hint at another expansion again in odd degrees but of a di¤erent basis, j nj = n 1 + 4 n+1 3 , j nj = n 1 + 12 n+1 3 + 36 n+2 5 j nj = n 1 + 28 n+1 3 + 240 n+2 5 + 576 n+3 7 . Those coe¢ cients are indeed positive integers, and they are derived in Section 5.1. The most detailed re…nement of n that we consider is the number of all k-tuples that intersect in the single number p, consist of h di¤erent subintervals, and have u left endpoints and v right endpoints. The number of such k-tuple of subintervals equals p 1 u 1 n p v 1 B(u; v; h)S(k; h)h!, (Section 4.3), where S (k; h) stands for the Stirling numbers of the second kind, and B(u; v; h) is the number of ways to select h elements from an u v matrix such that at least one element is chosen from each row and each column. At the same time, the numbers B(u; v; h) are the the connection coe¢ cients in the product formula (5) mn h = m X