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

Abstract

This note improves, in two respects, the results of §3.6 of my paper The hyper surface cross ratio. 1 There it is shown that the number c n 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 c n s =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.2 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 c n 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 8 ), 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. 352

On the enumeration of planar trees of hexagons

“…We let T(n) denote the number of rooted hexagon trees with n hexagons. This number was found by Motzkin [7] as the number of hexagonal dissections of polygons. The same number was independently found by the authors [1] in the enumeration of rooted fc-trees embeddable in fc-space, it being the number for k = 5.…”

Deux propriétés combinatoires des nombres de Schröder

“…Les nombres de Motzkin M"= £ [ ) C t sont aussi connus mais la biblio\2ij graphie les concernant est moins abondante (cf [6,4,5,17,24,28]). Ils énumèrent les mots de longueur n du langage algébrique M, dit de Motzkin, sur l'alphabet { x, x, y } dont l'équation est :…”

“…On retrouve les nombres de Motzkin dans rénumération de différentes familles d'arbres (zig-zag trees [4], branch-reduced plane trees [4], Y-branching plane trees [4], arbres de degré maximal 3 [17], arbres avec boucles [6], arbres sans sommet d'arité 1 [4]), des partitions d'un polygone [24], de chemins sous-diagonaux avec pas diagonal [6,22], de familles de graphes bipartis [6], de certains parenthésages [5], d'animaux dirigés à une seule source [45,14]. n / .s.…”

“…En effet si on code l'AFC en le parcourant suivant l'ordre préfixe et en écrivant une lettre x lorsqu'on rencontre une arête pour la première fois et une lettre x lorsqu'on rencontre une arête pour la deuxième fois sauf pour les arêtes pendantes coloriées 0 que l'on code y au lieu de xx, on obtient un mot [7,11,16,17,18,24,14,3,30]) et qui est aussi appelé parfois « Runyon number ». On obtient donc le corollaire suivant :…”

Vexillary Involutions are Enumerated by Motzkin Numbers

“…He also obtained a nice formula for 1234-avoiding involutions of length n that is ∑ n 2 i=0 n 2i (2i)!/((i + 1)!i! ), which are known as Motzkin numbers [36] beginning by 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, . .…”