Counting Interesting Elections

We provide an elementary proof of a formula for the number of northeast lattice paths that lie in a certain region of the plane. Equivalently, this formula counts the lattice points inside the Pitman–Stanley polytope of an n-tuple. Suppose that on election day a TV news network of questionable morality wants to increase their viewership as polling results come in. While the reporters cannot control the outcome of the election, they can control the order in which votes are reported to the public. If one candidate is ahead in the tally throughout the entire day, viewership will wane since it is clear that she will win the election. On the other hand, a more riveting broadcast occurs when one candidate is ahead at certain times and the other candidate is ahead at others. In fact, the network employs a group of psychologists and market analysts who have worked out certain margins they would like to achieve at certain points in the tally. The director of programming needs to know the number of ways this can be done. 1. The ballot problem We will work up to the general question by first examining the special (low ratings) case when one candidate has at least as many votes as the other throughout the tally. This is the classical “ballot problem”, in which candidate E and candidate N are competing for a public office. Candidate E wins the election with n votes. How many ways are there to report the votes so that at all times during the tally N is not ahead of E? We may represent the state of the tally at any moment by the pair (x, y), where the coordinates x and y count the votes received by E and N respectively. Then a tally consists of a sequence of points on the integer lattice in the plane made in steps of E = 〈1, 0〉 and N = 〈0, 1〉. Such a sequence is called a northeast lattice path. We say that the lattice path q is restricted by the lattice path p if no part of q lies directly above p. For example, Figure 1 shows two northeast lattice paths from (0, 0) to (n, n) that are restricted by the “staircase” p = ENEN · · ·EN , or, equivalently, that do not go above the line y = x. The ballot problem asks for the number Cn of these paths. (Note that if the tally ends at (n, m), we may uniquely continue it to a northeast lattice path ending at (n, n).) The ballot problem can be solved by constructing a simple recurrence. Let q be a northeast lattice path restricted by the staircase p. Consider the point on q where it first revisits the line y = x, and let i be the x-coordinate of this point. (This point exists since q ends at (n, n).) For the upper path in Figure 1, i = 3; for the lower path, i = 7. Date: October 30, 2009. 1 2 Lara Pudwell and Eric Rowland Figure 1. Two northeast lattice paths from (0, 0) to (7, 7) restricted by (EN). Notice that since q does not go above y = x and begins at (0, 0) its first step is E; further, its last step before reaching the point (i, i) is N . Therefore we may delete these steps to obtain a northeast lattice path from (1, 0) to (i, i − 1) that does not go above the line y = x − 1. There are Ci−1 ways to form such a path, and there are Cn−i ways to continue this path from (i, i) to (n, n), so we have that Cn = Σi=1Ci−1Cn−i. This we recognize as the familiar recurrence satisfied by the Catalan numbers Cn = ( 2n n ) /(n + 1) [8, Exercise 6.19(h)], so we simply check that the initial condition C0 = 1 agrees. 2. Notation and theorem We now consider a generalization of the ballot problem. Let LP(p) be the number of northeast lattice paths restricted by an arbitrary northeast lattice path p from (0, 0) to (n, m). The path p represents the network’s predetermined restrictions on the tally. It was known by MacMahon [5, p. 242] that the sum of LP(p) over all such paths is ∑ p LP(p) = (m + n)!(m + n + 1)! m!n!(m + 1)!(n + 1)! . However, we are interested in computing LP(p) for specific p. First we develop notation for lattice paths. It is possible to represent a northeast lattice path as a word on {E,N}, such as q = EENENNEENENNEN for the upper path in Figure 1. However, this representation is redundant, because the location of each E step determines the path uniquely. Therefore, we may represent a northeast lattice path by the sequence of heights qi of the path along each interval from x = i − 1 to x = i. For example, for the upper path in Figure 1 we have q = (0, 0, 1, 3, 3, 4, 6). This representation is always a nondecreasing tuple of integers, and it is our primary representation of lattice paths in this note. A lattice path q = (q1, q2, . . . , qn) is restricted by the lattice path p = (p1, p2, . . . , pn) precisely when q ≤ p componentwise, i.e., qi ≤ pi whenever 1 ≤ i ≤ n. Counting Interesting Elections 3 To write the main result, however, it turns out to be more natural to use still another representation of a northeast lattice path p — its difference sequence ∆p = (p1, p2 − p1, . . . , pn − pn−1). Let (v1, v2, . . . , vn) = v = ∆p. Since p is a northeast lattice path, the entries of v are nonnegative integers. The entry vi is the number of N steps taken along the line x = i− 1, so we can think of this representation as determining a path by the location of each N step. The operator ∆ has an inverse Σ, which produces the sequence of partial sums: p = Σv = (v1, v1 + v2, . . . , v1 + v2 + · · ·+ vn). The relationship between p and v = ∆p can be interpreted in another way. If v = (v1, v2, . . . , vn) is a tuple of nonnegative integers, the Pitman–Stanley polytope [7] defined by v is Πn(v) := { x ∈ R≥0 : Σx ≤ Σv componentwise } . Thus a tuple x = (x1, x2, . . . , xn) of nonnegative integers is a lattice point inside Πn(v) precisely when the northeast lattice path Σx is restricted by Σv. In other words, ∆ provides a bijection from the northeast lattice paths restricted by p to the lattice points in Πn(∆p). We now return to the question at hand: How many northeast lattice paths are restricted by the path p = (p1, p2, . . . , pn−1, pn)? Equivalently, how many lattice points lie inside Πn(∆p)? One answer to this question is the following determinant enumeration. Let A = (aij) be the n× n matrix with entries aij = ( pi+1 j−i+1 ) . Then the number of northeast lattice paths restricted by p is LP(p) = det A, as given by Kreweras [3] and Mohanty [6, Theorem 2.1]. This fact can be obtained from the triangular system of equations