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This note improves, in two respects, the results of §3.6 of my paper The hyper surface cross ratio. There it is shown that the number cn of independent hypersurface cross ratios that can be formed of 2n forms in n variables is 2 for n~ 2, 5 for w = 3, and 14 for n = 4. The proof employs the relations between cross ratios obtained by some simple permutations of the forms; let R be the set of these relations. It is remarked that the cross ratios of 2n — 1 forms in n variables, and of In — 1 forms in n — 1 variables, are connected by the same relations as the cross ratios of In forms in n variables, as far as these are consequences of the relations R, a "perhaps void restriction." We now prove that cn =C2n,n/(n+l)1 and that the restriction is in fact void, so that a complete knowledge of the relations between the cross ratios of 2n—l forms, of 2n forms, and of 2w+l forms in n variables Is obtained. The corresponding theorems for generalized intersections and one more variable are established at the same time. The same facts hold for a general class of function ratios, which includes hypersurface cross ratios and generalized intersections as very special cases. The number cn of independent function ratios has a simple combinatorial meaning, and appears also as the number of partitions of a polygon by non-intersecting diagonals into triangles, or of a cyclically arranged set into non-interlaced subsets, as the number of possibilities of never losing majority (in an election or a game), and as the number of different products of given terms in a given order, in a non-associative multiplication. For the combinatorial formula, seven proofs are given, six extended to generalizations.

bulletin (new series) of the american mathematical society/bulletin, new series, of the american mathematical society

Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products