Even and Odd Pairs of Lattice Paths with Multiple Intersections

This is a recount of the letters that I received from readers who continued where I left off by offering solutions to the problem. The problem is not that new. P. Erd6s, on reading the article, was quick to point out to me that Hungarian mathemat ic ians tackled it in the thirties: P. Veress proposing and G. Hajos solving it. In his letter to me I. Gessel (M.I.T.) has given a survey of the more recent history of the problem. Proofs were published by D. Kleitman (Studies in Applied Mathematics 54. (1975), also by his student D. J. Kwiatowski (Ph.D. Thesis, MIT, 1975). It also found its way into texts (Feller, Mohanty). In addition, I received solutions by A. Bondesen (Royal Danish School of Educational Studies, Copenhagen), K. Gr~nbaum (Roskilde Universitetscenter, Denmark), J. Hofbauer, jointly with N. Fulwick (Universit/it, Wien, Austria), D. Zeilberger (Drexel University, Philadelphia), and verbally from C. Pearce (Adelaide University), directly after reading the article. All solutions are based, with some variations, on the count of lattice paths, or equivalently (1,0) sequences. Figure 1 is used to illustrate the simplest version. It represents a two-dimensional coordinate lattice, or a network of streets running East and North. We consider paths of length 2n, beginning at O, proceeding in unit steps, heading East or North. It is clear that there are 22n ways in which a lattice point on the boundary AB can be reached. This gives the left-hand side of the identity. Counting in a different way, assume that the last crossing of a path with the NE line (OM on the diagram) is at K(k,k), which of course may coincide with O or M. It is easy to see that there are (2~) possible paths from O to K. Assuming for the moment that the number of ways the remaining 2n 2k steps, (avoiding OM) may be taken, is similarly (2n_-k2k), we obtain the desired right-hand side: ~=ot2k) (2n-~k) 9

The combinatorics of certain tuples of osculating lattice paths is studied, and a relationship with oscillating tableaux is obtained. The paths being considered have fixed start and end points on respectively the lower and right boundaries of a rectangle in the square lattice, each path can take only unit steps rightwards or upwards, and two different paths within a tuple are permitted to share lattice points, but not to cross or share lattice edges. Such path tuples correspond to configurations of the six-vertex model of statistical mechanics with appropriate boundary conditions, and they include cases which correspond to alternating sign matrices. Of primary interest here are path tuples with a fixed number $l$ of vacancies and osculations, where vacancies or osculations are points of the rectangle through which respectively no or two paths pass. It is shown that there exist natural bijections which map each such path tuple $P$ to a pair $(t,\eta)$, where $\eta$ is an oscillating tableau of length $l$ (i.e., a sequence of $l+1$ partitions, starting with the empty partition, in which the Young diagrams of successive partitions differ by a single square), and $t$ is a certain, compatible sequence of $l$ weakly increasing positive integers. Furthermore, each vacancy or osculation of $P$ corresponds to a partition in $\eta$ whose Young diagram is obtained from that of its predecessor by respectively the addition or deletion of a square. These bijections lead to enumeration formulae for tuples of osculating paths involving sums over oscillating tableaux.