In this paper is proved a formula for the number of simplicial neighbourly d-polytopes with d + 3 vertices, when d is odd

ALTSHULE, A

MCMULLEN, P

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263THE NUMBER OF SIMPLICIAL NEIGHBOURLYd-POLYTOPES WITH d+3 VERTICESA. ALTSHULER AND P. McMULLENAbstract. In this paper is proved a formula for the number of simplicial neigh-bourly d-polytopes with d + 3 vertices, when d is odd.§1. Introduction. A d-polytope P is k-neighbourly if every subset of k verticesof P is the set of vertices of a face of P. (We refer the reader to Griinbaum [1967,particularly §5.4 and §6.3] or McMullen-Shephard [1971, Chapter 3] for theterminology employed here.) The cyclic polytopes C(v, d) (v ^ d + 1) provideexamples of polytopes which are neighbourly (that is, n-neighbourly, n = [%d]);the only d-polytopes which are ^-neighbourly for k > n are the simplices. Theimportance of the simplicial neighbourly d-polytopes with v vertices lies in the fact,conjectured by Motzkin [1957] and proved by McMullen [1970], that, among allrf-polytopes with v vertices, they, and only they, have the maximal number of facesof each dimension.The cyclic polytopes, with their relatively simple structure, have been thoroughlyinvestigated. For the literature about cyclic polytopes prior to 1967, the readershould consult Griinbaum [1967]; for more recent results, we mention Altshuler[1971; 1973] and Shephard [1968] (see also McMullen-Shephard [1971, §2.3 (vi)]).Rather surprisingly, however, very little has been published about the structure ofthe other neighbourly polytopes; almost all the known information can be found inGriinbaum [1967, §7.2]. In particular, it is shown there that, if d is even, allneighbourly d-polytopes are simplicial, while if d is odd, for each v ^ d + 2 thereare neighbourly d-polytopes with v vertices which are not simplicial.Let bs(v, d, k) denote the number of simplicial ^-neighbourly d-polytopes withv vertices (so that cs(v, d) — bs{v, d, 1)). The known information about the mostinteresting case k = n (= [%d]) can be summarized as follows:If d = In, bs(v, d, n) = 1 if and only if d + 1 «S v < d + 3, and if d = In + 1,b,(v, d, n) = 1 if and only ifd+1 <r<rf + 2 (GrUnbaum [1967, §§7.2 and 7.3]);2>s(8,4,2) = 3 (Grunbaum-Sreedharan [1967]); and bs(9,4,2) = 23 (Altshuler-Steinberg [1973]).The main purpose of this paper is to prove:THEOREM 1. For n ^ 1,bs(2n + 4, In + 1, n) = 2^4(n + 2)h\n+2h oddwhere <f> is Euler's function.§2. Gale diagrams. Let P be a simplicial neighbourly (2« + l)-polytope with2n + 4 vertices. The combinatorial properties of P are faithfully reflected by thoseof its contracted standard Gale diagram P. (In matters of terminology concerning[MATHEMATIKA 20 (1973), 263-266]264 A. ALTSHULER AND P. McMULLENGale diagrams, we shall follow McMullen-Shephard [1971, §3.4].) P consists of2« + 4 points (counted according to multiplicity) distributed on the unit circle S inE2. P has an odd number, k say, of diameters (that is, diameters of S containingpoints of P), and points of P on adjacent diameters occur at opposite ends.The neighbourliness of P is equivalent to there being at least n + 1 points of Pon each side of every diameter (Griinbaum [1967, Exercise 7.3.7]). Thus, eachdiameter of P contains either one or two points of P, and (since the total number ofpoints is even), the number, Z say, of 2-diameters (that is, with two points of P) isalso odd. Moreover, since equal numbers of points of P on 2-diameters must lie oneither side of each 2-diameter, we see that adjacent 2-diameters carry their points ofP at opposite ends (compare the description of the Gale diagram of the uniqueneighbourly 2/j-polytope C(2K + 3, 2«) with 2« + 3 vertices). Finally, this impliesthat adjacent 2-diameters are separated by an even number of 1-diameters.The converse is clear: any Gale diagram with 2« + 4 points with an odd numberof 2-diameters, of which adjacent pairs are separated by an even number of1-diameters, is the Gale diagram of a simplicial neighbourly (2« + l)-polytope with2n + 4 vertices.§3. The proof of the theorem. We have reduced the proof of the theorem to theproblem of enumerating the Gale diagrams characterized in the last section. Witheach such Gale diagram, we associate a labelling of the vertices of a regular polygonwith an odd number of sides, as follows. If there are Z 2-diameters, let them benumbered D1 D, in cyclic order. Suppose that there are 2(/M, — 1) 1-diametersbetween Dt and Di+1 (i = 1, ..., /; Dt+l = Dt). Then we label the vertices of anZ-gon in cyclic order mu ..., mu noticing that, if the Gale diagram has k diametersin all,E m, = i £ 2(m, - 1) + Z = U(k - 0 + 21} = n + 2.il ilTo each isomorphism type of Gale diagram we are considering corresponds preciselyone equivalence class of such labelled regular polygons under orthogonal trans-formations, and conversely.It is therefore enough to count the number of essentially distinct ways of attachingto the vertices of regular Z-gons (Z = 1, 3, ...) positive integer labels totalling n + 2.But this problem is just a straight-forward application of the well-known theorem ofPolya [1937]. Indeed, the answer is implicit in the formula for 2)s (in case s is odd),given on page 169; it is the coefficient of x"+2 inxk 1 „ . _ xkS )_ i _- 1 9 t\ _ ~\/i _.2\(*-l)/2 ' ii.Z(l — ZX ) h odd 4/1which is the number given in the statement of the theorem.§4. Further results and conjectures. An immediate consequence of Theorem 1THE NUMBER OF SIMPLICIAL NEIGHBOURLY rf-POLYTOPES WITH d+3 VERTICES 265is thatlim bs(2n + 4,2n+ 1, n) = oo.n~*ooIt is reasonable to suppose that the following is also true.CONJECTURE 1. For every r > 3,lim bs(d + r, d, [%d]) = oo.d~* OOAnother result following easily from Theorem 1 is:THEOREM 2.lim b.(cs(2n + 4, In + 1)For, in Griinbaum [1967, §6.3] is given Perles' formula for cs(d + 3,d) (whichcan be proved by a method very similar to that of §3). In case d = 2« + 1, we havecs(2n + 4, In + 1) = 2" - (» + 2) + -(« +2)*|»+2k oddThus,cs(2n + 4, 2n + 1) > T - (n + 2) + _I_ .22n+1 = a, (say).n + 2On the other hand, since Z/.!n+2<K^) = n + 2, we easily see thatfes(2n + 4, 2« + 1, n) < 2[("(More careful estimates lead to the bound 2".) Sincelim ±1 = o,this proves the theorem.There is an obvious conjectural generalization of Theorem 2, along the lines ofConjecture 1. However, we shall propose instead two stronger generalizations; weshall state them as conjectures, although we do not wish to commit ourselves toofirmly to belief in them.CONJECTURE 2. For each fixed r 5= 3 and k > 1,lim b*-.oo bs(d + r,d, k)CONJECTURE 3. For each fixed r > 3 and k > 0,lim — ' = 0.In case r = 2, the appropriate limits are 1 and k/(k + 1) respectively.266 THE NUMBER OF SIMPLICIAL NEIGHBOURLY d-POLYTOPES WITH d+ 3 VERTICES§5. Remarks. Theorem 2 could be proved without using Theorem 1; from thedescription of star-diagrams in Griinbaum [1967, §6.2], together with the upper-bound theorem (McMullen [1970]), one can easily obtainbs(2n + 4,2n+ 1, n) < 2" + 1,which of course, leads to Theorem 2. However, Theorem 2 perhaps takes on a fullersignificance when viewed in the light of the consequence of Theorem 1 mentioned in §4.A brief remark about the origins of this paper is appropriate. The result ofTheorem 1 (in a somewhat different formulation) is due to the first author, while thesecond author provided the briefer proof presented here. We wish to thank ProfessorC. A. Rogers for putting us in contact, and thus making our collaboration possible.ReferencesA. Altshuler, " Polyhedral realization in R3 of triangulations of the torus and 2-manifolds in cyclic4-polytopes ", Discrete Maths., 1 (1971), 211-238., " Vertex figures of cyclic polytopes ", Notices Amer Math Soc. 19 (1972), A192.and L. Steinberg, " Neighborly 4-polytopes with 9 vertices ", J. Combinatorial Theory,Ser A, 15 (1973), 270-287.B. Grunbaum, Convex polytopes (Wiley and Sons, London, New York, Sydney, 1967).and V. P. Sreedharan, "An enumeration of simplicial 4-polytopes with 8 vertices ", /.Combinatorial Theory, 2 (1967), 437-465.P. McMullen, " The maximum numbers of faces of a convex polytope ", Mathematika, 17 (1970),179-184.and G. C. Shephard, Convex polytopes and the upper-bound conjecture, London Math. Soc.Lecture Notes Series, Vol. 3 (Cambridge University Press, 1971).T. S. Motzkin, " Comonotone curves and polyhedra ", Abstract 111, Bull. Amer. Math. Soc, 63(1957), 35.G. P61ya, " Combinatorische Anzahlbestimmungen fiir Gruppen, Graphen und chemische Ver-bindungen ", Acta Math., 68 (1937), 145-254.G. C. Shephard, "A theorem on cyclic polytopes ", Israel J. Math., 6 (1968), 368-372.University of the Negev, 52A25: CONVEX SETS AND GEOMETRICBeer-Sheva, Israel. INEQUALITIES; Convex polyhedra.University College London. Received on the 4th of September, 1972.

A theorem on cyclic polytopes &quot;,

Combinatorische Anzahlbestimmungen fiir Gruppen, Graphen und chemische Verbindungen &quot;,

Comonotone curves and polyhedra &quot;,

Convex polytopes (Wiley and Sons,

Neighborly 4-polytopes with 9 vertices &quot;,

Polyhedral realization in R 3 of triangulations of the torus and 2-manifolds in cyclic 4-polytopes &quot;,

The maximum numbers of faces of a convex polytope &quot;,

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NUMBER OF SIMPLICIAL NEIGHBORLY D-POLYTOPES WITH D+3 VERTICES

NUMBER OF SIMPLICIAL NEIGHBORLY D-POLYTOPES WITH D+3 VERTICES

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