Some Catalan Correspondences

One to one correspondences are established between planar rhyme schemes, planar partitions, connective relations and the associations of a product, which are all known to be enumerated by the Catalan numbers. The Catalan numbers c = [13; sequence 577] are well known as the number of ways of associating a product of n+1 terms [3, 4, 5,10,11,12]. Elementary ways of obtaining this result by consideration of the associated Polish forms are known [7,10,11,12]. There are a large number of other sets which are also enumerated by the Catalan numbers and many of these are readily put into one to one correspondence with one another [3, 4, 5, 10, 11, 12]. However other such enumerations often depend on showing the recurrence Cn = C0C,1_1 + C1Cn_2 + --• + Cn-lC0 and the one to one correspondences may not be readily apparent. Three such sets have recently occurred which are easily equivalent to one another, but are not obviously equivalent to the associations of n+ 1 terms, namely planar partitions [1], planar rhyme schemes [4, 6, 8, 9] and connective relations [8, 9,14]. In this note I present a one to one correspondence between these and the associations. Rogers has already observed the correspondence between planar rhyme schemes and connective relations in [9], where he gives a further correspondence between these and " similarity relations " which correspond to relations of nearness in a linearly ordered set. In [8], he gives the correspondence again and a recursive correspondence between connective relations and rooted planar trees (which are known to be enumerated by the Catalan numbers [4]). A planar partition of a regular «-gon in the plane is a partition of the n vertices as the components of a planar graph (undirected, without loops or multiple edges)