Applications of Narayana's formula

We adapt a formula due to T.V. Narayana to count ray paths from a source toa receiver according to the number of times the paths change direction. In the simplest case we show that Narayana's formula leads to a hook-sum formula for L(rn, n) posets. §l. Introduction and Motivation This paper arose in an attempt to solve a problem of counting certain ray paths in seismology. The key ingredient of the solution is a formula first discovered by Narayana [Nara 55]. To describe our problem, we consider profile of the earth with layers at equal intervals as shown in the diagram. Shock waves are generated by a source (S) at the 8th layer and we assume they take one unit of time to traverse a layer.l 1 In the terminology of seismology this means that all waves traversing the same number of layers are kinematic analogues [see CHP 89]. Australasian Journal of Combinatorics 1!( 1995), pp.47-57 Each time a wave is reflected from the boundary of a sel:sn:lO:gn:tpJtllc record is reduced. To interpret this its contribution to the it is of interest to know the proportion of waves which arrive at the receiver (R) at the rth layer after having traversed n + m and having been reflected R. times where R. traversed in a downward direction and m is the number traversed in an upward direction. We say that n + m is the length of the wave. It is clear that we can represent each wave of n+m as a X t X 2 ··· X n +m of D's (for downward) and U's (1tpward) which is subject to the condition that the difference between the number of U's and the number of D's is at most s for each substring of the form X 1X 2 ··· Xi. We need this surface condition to ensure that the wave does not disappear from the Oth layer. However we make no restriction on how deep the wave may penetrate the earth. In the case r s 0 (receiver and source are at the surface of the earth) we have n m and satisfying the above condition are often called balanced. It is well-known that the total number of balanced strings of length 2n is the nth Catalan number Cn (2:). For example, if r = 0 and n = m = 4, one wave is reflected once, six waves are reflected three times, six waves are reflected five times and one wave is reflected seven times, for a total of C4 = 14, illustrated in the following diagram. More generally, let T~~(s) be the number of strings of length n + m which contain k pairs of the form DU and i pairs of the form U D (representing a wave which is reflected i + k times) generated by a source layers below the surface. We write T~k = T::/k(O). Clearly, Ii kl :::; 1. In particular, if r = s = 0 and n = m we have the identity, By a formula of Narayana [Nara 55, Nara 59], T:::-1 ~ (~) (k:l)' In the above example, Tt? = 1, Ttl = 6, Tt; = 6, Ttl = 1 as expected.