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

MCMULLEN, P

English

UCL Discovery

26THE NUMBER OF NEIGHBOURLY ^-POLYTOPES WITHd+3 VERTICESP. McMULLENAbstract. In this paper is proved a formula for the number of neighbourly d-polytopes with d+3 vertices, when d is odd.§1. Introduction. In an earlier paper (Altshuler-McMullen [1973]), a formulawas found for the number of simplicial neighbourly d-polytopes with d + 3 vertices,in the case when d is odd. (We refer the reader to that paper for background infor-mation on neighbourly polytopes relating to the problem considered here, and toGriinbaum [1967] or McMullen-Shephard [1971] for more general information onpolytopes.) In this paper, we shall solve the more general problem of enumerating allthe neighbourly d-polytopes with d + 3 vertices. Specifically, letting b(v, d, k) denotethe number of combinatorial types of ^-neighbourly d-polytopes with v vertices, weshall prove:THEOREM. For n ^ 1,b(2n + 4,2n + 1, n) = i{(5 + (- 1)") 3[(n+1)/2] + 6}+ .. * Z 4(n + 2)/i|»+2h oddwhere cj> is Euler's function.§2. Gale diagrams. Let P be a neighbourly (2n + l)-polytope with 2n + 4vertices. Since the combinatorial properties of P are faithfully reflected by those ofits Gale diagram P (see Griinbaum [1967, §§5.4 and 6.3] and McMullen-Shephard[ 1971, §3.4]), it will be enough for our purposes to enumerate the isomorphism cla s sesof the corresponding Gale diagrams.We first remind the reader of the characterization of the Gale diagrams of simplicialneighbourly (2m + l)-polytopes with 2m + 4 vertices (m > 1). Since we shall needto vary m (and even allow, formally, m = 0 or — 1), we introduce the followingterminology. The points of the diagrams Q (say) which we consider occur on, or atthe centre of the unit circle S in E2; a diameter of Q is a diameter of S containingpoints of Q on S. We say Q is balanced if each diameter L of Q contains one or twopoints only, and the remaining points of Q (2m + 3 or 2m + 2 in number) are equallydistributed on each side of L (that is, at least m + 1 on each side). Then the Galediagrams of simplicial neighbourly (2m + l)-polytopes with 2m + 4 vertices are thebalanced diagrams with 2m + 4 points, with no points at the centre of S, and suchthat no diameter carries points at opposite ends. In contracted form, adjacent dia-meters carry their points (one or two in number) at opposite ends. There are an oddnumber of diameters which carry two points, and adjacent diameters carrying twopoints are separated by an even number of diameters which carry one point. Indistended form, we replace each diameter carrying two points by a pair of adjacent[MATHEMATIKA 21 (1974), 26-31]THE NUMBER OF NEIGHBOURLY J-POLYTOPES WITH d+ 3 VERTICES 27Contracted DistendedFig. 1diameters carrying one point at the same end. (Figure 1 illustrates this situation,with m = 2.) We may thus group the 2m + 4 diameters (each carrying a single point)in adjacent pairs, k (which is odd) of which carry their points at the same end, andthe remaining m + 2 — k of which carry their points at opposite ends.Let us now consider the general situation of the Gale diagram of a not necessarilysimplicial neighbourly (2« + l)-polytope with 2n + 4 vertices. The fact that thediagram is balanced implies, as before, that each of its diameters contains at mosttwo points. But now a diameter may have a point at each end, or a point at one end,and one at the centre of the circle S.The latter case is easy to deal with. The Gale diagram is that of a pyramid, whosebasis, being a neighbourly 2«-polytope with 2n + 3 vertices, is the cyclic polytopeC(2n + 3, 2n).Another case is also easily added to our list, the case when every diameter has apoint at each end. Thus there are n + 2 diameters in all. This diagram is also essenti-ally unique.From now on, then, we shall suppose our Gale diagram P has no point at thecentre of S, and not all the diameters of P have a point at each end. Let us alsosuppose that P is in distended form, so that each diameter carrying two points atthe same end is replaced by a pair of adjacent diameters, carrying a single point at thesame end.We first observe that, if we now remove a diameter containing a point at each end,the new diagram is still balanced. So, if there are, say, p of these diameters, and weremove them all, we are left with a balanced diagram containing 2(n — p + 2)points, each point on a distinct diameter. Using the notation introduced above,let us suppose there are k (odd) pairs of adjacent diameters carrying their points atthe same end, say K/(i = 0,l;j = 1, ..., k), and / = (n — p + 2) — k pairs carryingtheir points at opposite ends, say L'j(i = 0, 1; j = 1, ..., /). Moreover, let us supposethat, for j = 1, ..., k, there are /,- pairs L's separating K/ and Kj+ x (K'k+1 = K^).28 P. MCMULLENWe now attempt to replace the diameters carrying points at each end. As far asthe diameters of type K/ or L'j are concerned, inserting such a diameter does notaffect the property of being balanced. So, we are only concerned to ensure that equalnumbers of the 2(« — p + 2) original points lie on each side of the new diameters.Clearly, we may separate two diameters Kj° and K/. But as we rotate the new diameteraway from this initial separating position, we see that the difference between thenumbers of points on each side is alternately 2 or 0, changing each time a diameterK/ or L'j is crossed. In other words, the allowable positions to insert the new dia-meters are separating a pair {Kf, K/} or {L°s, Lxs}; we may not separate adjacentdiameters of different pairs.Suppose, then, that for j = 1, ..., k,q} - 1 new diameters separate Kf and K/,and for j = 1, ...,/, rt — 1 separate L°j and Ll}. Changing the notation slightly, wesee that to our diagram corresponds the sequence of positive integers,(<Zl> rll> •••>''lli' #2> r2\i ••••> rl\%i ••••> Qk> rkl> •••' rklk)>whereConversely, for any odd k, such a sequence (with varying q}, lj and rjs) gives rise to abalanced Gale diagram with 2n + 4 points.So, to find b(2n + 4, 2n + 1, n), we must enumerate these sequences. However,we must bear in mind that the sequences should really be regarded as cyclic, and sotwo such sequences must be identified if one can be obtained from the other by cyclicpermutation or reversal of order, as long as the special roles of qu ...,qk are preserved.We shall solve this enumeration problem in the next section.§3. The proof of the theorem. We solve the enumeration problem of §2 by meansof Burnside's Lemma. That is, for each k = 1, 3, ... we count the number of arrange-ments invariant under each possible symmetry, and average this over this wholegroup of symmetries, which is, in this case, the dihedral group Dk.The symmetries are of two kinds: k reflexions, and k rotations (there are <f>(li)of order exactly h, for each divisor h of k, where <j> is Euler's function). We firstconsider the k reflexions. The invariant arrangements are all similar, so it is enoughto consider a typical example, that which leaves qt fixed, and interchanges the pairs{<72>4fc}> {<?3><7*-iK •••> and so on. So, we must also have /t = lk, l2 = lk_u ..., andso on, and, for; = 1, ...,\{k + 1), s = 1, ..., lj,rjs — rk-j + 2,lj-s+l-In the case j = %(k + 1), there are two cases, according as l(k+r)/2 is even or odd.In the latter case, the term r(k+1)/2; ilj+ ,)/2 is also fixed by the reflexion. In the formercase, there is no such invariant term; to treat the two cases together, we shall herereplace /(M-,)/2 by l + /(t+1)/2, relabel r(t+1)/2jS by r(k+n/2tS+1 for s ;* \l(k+i)i2, andintroduce the term r(jt+1)/2, m + t = 0, with m = %l(k+i)/2.We conclude, therefore, that for reflexion terms, we must enumerate the sequences3l> rll> •••» rUi'i2' ••THE NUMBER OF NEIGHBOURLY rf-POLYTOPES WITH d+3 VERTICES 29where all terms (except possibly the last) are positive, and9i + 2(ru + ... +r(k+ 1)/2.t-i) + ^k+iy2>t = n + 2.We first need a lemma.LEMMA. The number of sequences (xt, ...,xv) of positive integers with sum m is(::.')•The proof is by induction on the length v of the sequence; it is easy, and is thereforeleft to the reader.So, for each m < [•£(« + 1)], we have ("r/) possible sequences (xu ...,*„) oflength v. (To simplify certain expressions below, we adopt the special convention(ll) = 1.) We can choose %(k — 1) of the xt to play the roles of q2, ••-,q(k+i)/2 in(i(t-i)) ways; the remaining xfs play the roles of rlu ..., r(k+1y2tt-i- We must nowaccount for*?! and rik+1y2,t from the n + 2 — m points left over; sinceqt is positive,this can be done in n + 2 — 2m ways. Hence, each reflexion term contributes a totalof(K — 1)12It follows that the total contribution from all reflexion terms (for all k) isIk odd Z Z (i. + 2 - 2») ( .. \y2)(m~!)}m«[(»+l)/2]tS0 \ (K — l)/2 / \ V — 1 ))m«C("+l)/2] v&0 \ \ V — 1 / k odd \ (k — l)/2/j= i(» + 2) + (« + 2)2-2/n)3m"1- 1- 23-13t("+3)/2] - [(« + 3)/2] 3[(n(3 - I)2{(« + 3 - 2[(n +l)/2]) 3[(n+1)/2] - 1}30 P. MCMULLENThe rotation terms are rather easier. For each of the <f>(h) rotations of order h inDk, we must count the sequenceswhose sum is (n + 2)jh (taking, as before, a typical sequence). Of course, h must be acommon divisor of k and n + 2. The number of such sequences of total length v is/ v - 1 \ /((« + 2)lh) - 1 \ = k I v \/(n + 2)/h\\{k/h)-l)\ v-l ) n + 2\k!h)\ v )'since the role of qx is already assigned, and we can choose any (k/h) — 1 out of theremaining v — 1 to play the roles of q2, • ••,<Z/t/;,.Thus the total rotation term (again, for all k) is£^/codd IK2(n + 2) koddV /(« + » l({n + 2)lh)-klh\v^o \ k/h ) \ ((« + 2)/A) - » /2) A|n + 2^ Yoddh oddZ 0(A)i(3("+2)/h — 1).2(n + 2v ' H oddIf we now collect these terms together, and recall the two additional exceptionalcases, we obtain the formula given in the theorem.§4. Remarks. In the special case n = 1 (so d = 3), we obtainb(6, 3, 1) = 7.Of course, this is just c(6, 3), since every 3-polytope is 1-neighbourly.In Altshuler-McMullen [1973], two conjectures were made about the behaviourof bs(v, d, k), the number of simplicial ^-neighbourly d-polytopes with v vertices.We wish here to propose as problems:CONJECTURE 1. For each fixed r ^ 3 and k ^ 1,b(d + r, d, k + 1) _d->a> b(d + r, d, k)CONJECTURE 2. For each fixed r > 3 and k~2* 1,b(d+r,d,[W]-k+l)*-» i(d + r, d, [id] - k)THE NUMBER OF NEIGHBOURLY rf-POLYTOPES WITH d+ 3 VERTICES 31Sinceb(d + 2, d, k) = [i(d -2k + 2)2],the appropriate limit for r = 2 in the first problem is 1. In the second problem, whenr = 2, the ratio takes the valueor k + 2'according as d is even or odd.ReferencesA. Altshuler and P. McMullen, 1973. " The number of simplicial neighbourly rf-polytopes withd+3 vertices," Mathematika, 20 (1973), 263-266.B. Griinbaum, 1967. Convex polytopes (Wiley and Sons, London-New York-Sydney, 1967).P. McMullen and G. C. Shephard, 1971. Convex polytopes and the upper bound conjecture, LondonMath Soc. Lecture Notes Series, Vol. 3 (Cambridge University Press, 1971).University College London. 52A25: CONVEX SETS AND GEOMETRICINEQUALITIES; Convexpolyhedra.Received on the 4th of September, 1972.

NUMBER OF NEIGHBORLY D-POLYTOPES WITH D+3 VERTICES

Abstract

Similar works

Full text

Available Versions

UCL Discovery