The Partitionability ConjectureArt M. Duval, Caroline J. Klivans, and Jeremy L. Martin

Communicated by Benjamin Braun

In 1979 Richard Stanley made the following con-jecture: Every Cohen-Macaulay simplicial complexis partitionable. Motivated by questions in thetheory of face numbers of simplicial complexes,the Partitionability Conjecture sought to connect a

purely combinatorial condition (partitionability) with analgebraic condition (Cohen-Macaulayness). The algebraiccombinatorics community widely believed the conjectureto be true, especially in light of related stronger conjec-tures and weaker partial results. Nevertheless, in a 2016paper [DGKM16], the three of us (Art, Carly, and Jeremy),together with Jeremy’s graduate student Bennet Goeck-ner, constructed an explicit counterexample. Here wetell the story of the significance and motivation behindthe Partitionability Conjecture and its resolution. Thekey mathematical ingredients include relative simplicialcomplexes, nonshellable balls, and a surprise appearanceby the pigeonhole principle. More broadly, the narrativeof the Partitionability Conjecture highlights a generaltheme of modern algebraic combinatorics: to understanddiscrete structures through algebraic, geometric, andtopological lenses.

History and MotivationAs basic discrete building blocks, simplicial complexesarise throughout mathematics, whether as surfacemeshes, simplicial polytopes, or abstract models ofmultiway relations. An abstract simplicial complex Δ issimply a family of subsets of a finite vertex set 𝑉 thatis closed under taking subsets: if 𝜎 ∈ Δ and 𝜏 ⊆ 𝜎,then 𝜏 ∈ Δ. The elements of Δ, which are called faces orsimplices, admit a natural partial order by containment,and we typically regard a simplicial complex as equivalentto its face poset (partially ordered set of faces). Themaximal faces are called facets. A simplicial complex canbe realized topologically by representing each face 𝜎 bya geometric simplex of dimension |𝜎| − 1, as in Figure 1.

Art M. Duval is professor of mathematics at the University ofTexas at El Paso. His e-mail address is aduval@utep.edu.

Caroline J. Klivans is lecturer in applied mathematics and com-puter science at Brown University and associate director of theInstitute for Computational and Experimental Research in Mathe-matics (ICERM). Her e-mail address is klivans@brown.edu.

Jeremy L. Martin is professor of mathematics at the University ofKansas. His e-mail address is jlmartin@ku.edu. His work is par-tially supported by Simons Foundation Collaboration Grant No.315347.

For permission to reprint this article, please contact:reprint-permission@ams.org.DOI: http://dx.doi.org/10.1090/noti1475

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Figure 1. A simplicial complex and its face poset.

For this reason, the standard convention is to write thedimension of Δ (that is, the largest dimension of a faceof Δ) as 𝑑 − 1. Here we are concerned with the problemof decomposing the face poset into intervals, i.e., subsetsof the form [𝜎,𝜏] ∶= {𝜌 ∈ Δ | 𝜎 ⊆ 𝜌 ⊆ 𝜏}.

The 𝑓-vector of a complex records the face numbers,dimension by dimension. Specifically, the 𝑓-vector is𝑓(Δ) = (𝑓−1, 𝑓0,… , 𝑓𝑑−1), where 𝑓𝑖 is the number of 𝑖-dimensional faces in Δ. Understanding the 𝑓-vectorsof various classes of complexes is a major area ofalgebraic combinatorics; the standard source is [Sta96].The Partitionability Conjecture began as an attempt togain a combinatorial understanding of the face numbersof Cohen-Macaulay complexes.

The classical problem in this subject is to characterizethe 𝑓-vectors of convex polytopes. For example, (1, 5, 9, 6)is realized by the polytope in Figure 1, and it is not hardto convince yourself that (1, 5, 9, 87) is impossible, butwhat about (1, 15, 36, 12)? What can be said about thiscombinatorial invariant of inherently geometric objects?

February 2017 Notices of the AMS 117

This problem dates back at least to Euler and remainsunsolved in general, though many special cases areunderstood.

It often turns out to be more convenient to work witha linear transformation of the 𝑓-vector called the ℎ-vector,ℎ(Δ) = (ℎ0,… , ℎ𝑑), defined by the identity

𝑑∑𝑖=0

ℎ𝑖𝑥𝑑−𝑖 =𝑑∑𝑖=0

𝑓𝑖−1(𝑥 − 1)𝑑−𝑖.

The formula can easily be inverted to express the 𝑓-vectorin terms of the ℎ-vector, so the two carry the sameinformation. The ℎ-vector is a key thread in our narrative,andwewill see that it carries simultaneous interpretationsin algebra, combinatorics, and topology.Many relations onthe face numbers, particularly those arising in algebraiccontexts, can be expressed naturally in terms of theℎ-vector. A spectacular example is the Dehn-Sommervilleequations, which state thatℎ𝑖 = ℎ𝑑−𝑖 for every simplicial𝑑-polytope. (This symmetry reflects Poincaré duality on theassociated toric variety.) What do the ℎ-numbers count?In general they need not even be positive, but for certainspecial classes such as shellable simplicial complexesthey admit an elementary combinatorial interpretation. Ashellable complex can be assembled facet by facet so thatthe new faces added at each step form an interval in theface poset; the number ℎ𝑖 counts the intervals of height𝑑 − 𝑖. The boundary complexes of convex polytopes areshellable, as proven by a beautiful geometric argumentof Heinz Bruggesser and Peter Mani in 1970. Shellabilityof polytopes is an essential ingredient in a cornerstoneof the theory of face numbers, Peter McMullen’s upperbound theorem, which describes an upper bound thatsimultaneously maximizes all entries of the ℎ-vectors ofpolytopes.

Mel Hochster isoften quoted assaying that “lifeis really worthliving” in a

Cohen-Macaulayring.

A condition weakerthan shellability is par-titionability, which firstappears in the thesesof Michael O. Ball andJ. Scott Provan. Ball hadbeen working on networkreliability, and Provan ondiameters of polytopes. Apartitionable complex canbe decomposed into dis-joint intervals, each onetopped by a facet. Thatis, a partitioning matcheseach facet to one of itssubfaces (possibly itself),

namely, the minimum element of the corresponding in-terval. Unlike a shelling, there is no restriction on howthe intervals of a partitioning fit together. On the otherhand,when𝑋 is partitionable, the number ℎ𝑖 counts the in-tervals of height 𝑑−𝑖, just as for a shellable complex. (SeeFigure 2.) Hence the ℎ-vectors of partitionable complexeshave a simple combinatorial interpretation.

Constructible and Cohen-Macaulay simplicial com-plexes also have nonnegative ℎ-vectors. Constructibility is

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Figure 2. A shellable simplicial complex with ℎ-vector(1, 2, 2, 1) and the corresponding partitioning of theface poset into intervals.

a purely combinatorial recursive condition, while Cohen-Macaulayness arises in commutative algebra. Cohen-Macaulay rings have long enjoyed great importancein algebra and algebraic geometry; Mel Hochster is of-ten quoted as saying that “life is really worth living”in a Cohen-Macaulay ring. Their significance in com-binatorics, via what is now known as Stanley-Reisnertheory, was established by Hochster, Gerald Reisner, andRichard Stanley in the early 1970s. In this theory com-binatorial questions about a complex are translated toalgebraic questions about a ring. Specifically, the Stanley-Reisner ring (or face ring) of a simplicial complex Δ onthe vertex set {1, 2,… ,𝑛} (over a field 𝕜) is defined as𝕜[Δ] ∶= 𝕜[𝑥1,… , 𝑥𝑛]/𝐼Δ, where 𝐼Δ is the monomial idealgenerated by nonfaces of Δ. The construction is bijective:every quotient of a polynomial ring by a square-freemonomial ideal gives rise to a simplicial complex. Thetwo sides of the correspondence are tightly linked: forinstance, the ℎ-vector of Δ corresponds naturally to theHilbert series of 𝕜[Δ] via the formula

∑𝑖≥0

(dim𝕜[Δ]𝑖)𝑡𝑖 =∑𝑗 ℎ𝑗(Δ)𝑡𝑗

(1 − 𝑡)𝑑 ,

where 𝕜[Δ]𝑖 denotes the 𝑖th graded piece of 𝕜[Δ].Two fundamental algebraic invariants of a commuta-

tive ring 𝑅 are its (Krull) dimension and its depth. In all

118 Notices of the AMS Volume 64, Number 2

cases dim𝑅 ≥ depth𝑅, and if equality holds, then thering is Cohen-Macaulay. Likewise, a simplicial complexis Cohen-Macaulay if its Stanley-Reisner ring is Cohen-Macaulay. In the setting of Stanley-Reisner theory, thedimension of 𝕜[Δ] is simply 𝑑, but depth is a subtlerinvariant of Δ, so it is not easy to translate this defini-tion from algebra to combinatorics. Fortunately, Reisnerfound an equivalent criterion for Cohen-Macaulayness interms of simplicial homology: Δ is Cohen-Macaulay if andonly if for every face 𝜎, the subcomplex linkΔ(𝜎) = {𝜏 ∈Δ | 𝜎∩𝜏 = ∅, 𝜎∪𝜏 ∈ Δ} has the homology type (over𝕜) of a wedge of ((𝑑 − 1) − dim𝜎)-spheres. Reisner’scriterion is the working definition of the Cohen-Macaulayproperty for many combinatorialists.

The Cohen-Macaulay property provided the algebraicbridge between topology and combinatorics in Stanley’scelebrated extension of McMullen’s upper bound theo-rem from polytopes to simplicial spheres. Specifically,Cohen-Macaulayness played the same role in bound-ing the ℎ-vectors of spheres that shellability did forconvex polytopes. Furthermore, the hierarchy of strictimplications

shellable ⟹ constructible ⟹ Cohen-Macaulay

was well known since Hochster’s 1973 work on facerings. (Of these properties, only Cohen-Macaulayness istopological, as proven by James Munkres in 1984.) On theother hand, Stanley proved in 1977 that the ℎ-vectors ofCohen-Macaulay, shellable, and constructible complexesare precisely the same. The Partitionability Conjecturesought to explain this equality: perhaps partitionabilitysat at the base of the hierarchy above.

Conjecture 1 (The Partitionability Conjecture). EveryCohen-Macaulay simplicial complex is partitionable.

The Partitionability Conjecture is one of many in-terrelated conjectures about the structure of simplicialcomplexes. Adriano Garsia made the same conjecture in1980 in the more restricted setting of order complexesof Cohen-Macaulay posets. Motivated by the theory ofalgebraic shifting, Gil Kalai conjectured that every simpli-cial complex admits a more general decomposition intointervals. In 1993 Stanley conjectured that every 𝑘-acycliccomplex—one for which linkΔ(𝜎) is acyclic for everyface 𝜎 of dimension less than 𝑘—admits a partitioninginto intervals of rank at least 𝑘 + 1, with each intervaltopped by a face of dimension at least depthΔ. Eachof Kalai’s and Stanley’s conjectures, if true, would im-ply the Partitionability Conjecture. Meanwhile, MasahiroHachimori proved in 2000 that a certain strengtheningof constructibility does in fact imply partitionability. ThePartionability Conjecture was widely believed to be truewithin the combinatorics community, and the worksof Garsia, Kalai, Stanley, and Hachimori provided manytantalizing approaches, both combinatorial and algebraic.

The Partitionability Conjecture gained additional signif-icance in thestudyofStanley depth, apurely combinatorialcounterpart to the depth of a graded module over apolynomial ring, introduced by Stanley in 1982. This

invariant has attracted substantial attention in combina-torial commutative algebra over the last twenty years;the article [PSFTY09] by Mohammed Pournaki, Seyed A.Seyed Fakhari, Massoud Tousi, and Siamak Yassemi in theNotices is an excellent introduction. The focal point of thisarea has been Stanley’s conjecture that the Stanley depthof every module is an upper bound for its (algebraic)depth. Jürgen Herzog, Ali Soleyman Jahan, and Yassemiproved in 2008 that Stanley’s Depth Conjecture impliesthe Partitionability Conjecture.

On the other hand, as with so many areas of mathe-matics, combinatorics is full of surprises, and the studyof simplicial complexes is rife with counterintuitive exam-ples. Shellability does not depend only on the topologyof the underlying space: Mary Ellen Rudin famously con-structed a nonshellable triangulation of the 3-dimensionalball in 1955. There are also nonshellable 3-spheres; thefirst was constructed in 1991 by William Lickorish. Itcan get even more wild: obstructions to shellability andconstructibility can take the form of nontrivial knotsembedded on a sphere. So perhaps the PartitionabilityConjecture would also turn out to be false.

Building a CounterexampleWe started by trying to prove the Partitionability Con-jecture, not to disprove it. Our previous joint work onsimplicial and cellular trees and follow-up work of CarlyandOlivier Bernardi had revealed unexpected connectionsto the Cohen-Macaulay condition and to Stanley’s conjec-tures about decompositions of simplicial complexes. Atfirst, we had hoped to use the theory of trees to approachpartitionability. However, we eventually arrived at twoturning points, key realizations that persuaded us to lookfor a counterexample instead of a proof.

When we first began to investigate partitionability, wewanted to experiment with a Cohen-Macaulay complexthat is not shellable, because as we have seen, everyshelling induces a partitioning. However, many Cohen-Macaulay complexes that arise in combinatorics, such asconvex simplicial spheres and order complexes of certainposets, are naturally shellable. Rudin’s nonshellable 3-ballwould have met our requirements, but we chose the oneconstructed by Günter Ziegler in 1998 [Zie98], since it hasfewer faces and is thus easier to work with. It has 𝑓-vector(1, 10, 38, 50, 21) and is known to be partitionable.

Almost any approach to working with decompositionsof simplicial complexes also requiresworkingwith relativesimplicial complexes. A relative simplicial complex 𝑄 onvertex set 𝑉 is a family of subsets of 𝑉 that is convexwith respect to the natural partial order by inclusion:if 𝜌 and 𝜏 are faces of 𝑄 with 𝜌 ⊆ 𝜎 ⊆ 𝜏, then 𝜎 isa face of 𝑄 as well. Equivalently, the face poset of 𝑄is of the form 𝑋\𝐴, where 𝑋 is a simplicial complexand 𝐴 is a subcomplex (see Figure 3). The concept ofa pair of spaces is familiar from algebraic topology;in particular, 𝑄 can be regarded as a combinatorialmodel of the quotient space 𝑋/𝐴. The combinatorics ofsimplicial complexes, including Cohen-Macaulayness andpartitionability, carries over well to the relative setting.

February 2017 Notices of the AMS 119

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Figure 3. A simplicial complex as the union of a subcomplex and a relative complex.

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Figure 4. The nonpartitionable Cohen-Macaulay relative simplicial complex 𝑄5 shows that thePartitionability Conjecture is false for relative simplicial complexes. The labeling of verticesfollows Ziegler [Zie98].

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Figure 5. Gluing two isomorphic copies of 𝑋 along 𝐴. The resulting complex is Cohen-Macaulay forappropriate 𝑋 and 𝐴 but may still be partitionable, as in this example.

Furthermore, the ability to remove subcomplexes makesit the right setting to study decompositions.

Our first turning point arose from one of our ex-periments on Ziegler’s 3-ball. We used a simple greedyalgorithm to remove intervals one at a time, producing asequence of smaller and smaller relative complexes, eachof which remains Cohen-Macaulay (by a Mayer-Vietoris

argument). In one trial, the final output was the relativecomplex 𝑄5, whose face poset is shown in Figure 4. Itis not hard to check directly that 𝑄5 is nonpartition-able. Therefore, the Partitionability Conjecture is false forrelative simplicial complexes.

We next wanted to find a general method for turninga relative counterexample 𝑄 = (𝑋,𝐴), such as 𝑄5, into

120 Notices of the AMS Volume 64, Number 2

a nonrelative counterexample. Our first idea was toconstruct a complex 𝐶2 by gluing two isomorphic copiesof 𝑋 together along the common subcomplex 𝐴, asin Figure 5; this complex is Cohen-Macaulay (again, by aMayer-Vietoris argument). The single extra copy of𝑄 doesnot, however, create an obstruction to partitionability,since it may be possible to partition 𝐶2 by pairing facesof 𝐴 with facets in different copies of 𝑄.

But now the pigeonhole principle comes into play.Construct a Cohen-Macaulay complex 𝐶𝑁 by gluing 𝑁copies of 𝑋 together along their common subcomplex 𝐴.Suppose that 𝑁 is greater than the total number of facesof 𝐴. Then, in every partitioning of 𝐶𝑁, there must beat least one copy of 𝑄 whose facets are matched withfaces in that same copy. That is, 𝑄 is partitionable! Thiscontradiction implies that 𝐶𝑁 cannot be partitionable.Here is a formal statement:Theorem 2. Let 𝑄 = (𝑋,𝐴) be a relative complex suchthat

(i) 𝑋 and 𝐴 are Cohen-Macaulay,(ii) 𝐴 has codimension at most 1,(iii) every minimal face of 𝑄 is a vertex, and(iv) 𝑄 is not partitionable.

Let 𝑘 be the total number of faces of 𝐴, let 𝑁 > 𝑘, and let𝐶 = 𝐶𝑁 be the simplicial complex constructed from 𝑁 dis-joint copies of 𝑋 identified along the subcomplex 𝐴. Then𝐶 is Cohen-Macaulay and not partitionable.

Condition (iii) is a technical requirement to ensure that𝐶 is a simplicial complex, not merely a cell complex. Forinstance, the relative counterexample 𝑄5 does not satisfycondition (iii).

This flexibilitystruck us as suspiciously

strong.

It was not clear that The-orem 2 would produce anactual counterexample: findinga relative complex satisfyingthe conditions of the theoremmight be just as difficult asproving or disproving Conjec-ture 1 through other means.But by considering the specialcase that 𝐴 is a single facet, werealized that the conjecture implied a stronger versionof itself: If 𝑋 is a Cohen-Macaulay complex, then 𝑋 hasa partitioning including the interval [∅,𝜎] for any facet𝜎. This flexibility struck us as suspiciously strong: for ex-ample, the corresponding statement fails for shellability.Here was our second turning point.

Now we were determined to find a complex satisfy-ing the conditions of Theorem 2. Once again, Ziegler’sball 𝑍 provided the answer. Consider the subcomplex𝐵 consisting of all faces supported on the vertex set{0, 2, 3, 4, 6, 7, 8}, so that the minimal faces of the relativecomplex 𝑄 = 𝑍/𝐵 are vertices 1, 5, and 9. We provedthat 𝐵 is Cohen-Macaulay and that 𝑄 is not partitionable.These were exactly the ingredients we needed to constructa counterexample!

We can describe 𝑄 most simply as the relative com-plex (𝑋,𝐴), where 𝑋 is the smallest simplicial complexcontaining 𝑄. In fact 𝑋 is a 3-ball with 10 vertices and 14

Figure 6. A model of 𝑋, the smallest simplicialcomplex containing 𝑄. The subcomplex 𝐴 consists ofthe five triangles without green vertices.

tetrahedra (see Figure 6), and 𝐴 is a topological disk onthe boundary of 𝑋, with 𝑓-vector (1, 7, 11, 5). Therefore,gluing (1 + 7 + 11 + 5) + 1 = 25 copies of 𝑋 along 𝐴produces a counterexample 𝐶25 to the PartitionabilityConjecture. The complex 𝐶25 is Cohen-Macaulay (in fact,constructible) for the usual Mayer-Vietoris reasons, but itis nonpartitionable by the pigeonhole argument.

In fact, the argument that 𝑄 is nonpartitionable re-vealed that the full power of Theorem 2 was actually notnecessary. Gluing just three copies of 𝑋 together along𝐴 gives a nonpartitionable complex 𝐶3. The complex 𝐶3

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1″

5″ 9″

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Figure 7. A partial representation of thecounterexample 𝐶3: three copies of the 3-ball 𝑋 gluedtogether along the 2-ball 𝐴. The figure shows all 16vertices and 71 edges, but not all of the 98 triangles,nor any of the 42 tetrahedra.

February 2017 Notices of the AMS 121

is the smallest counterexample we know: its 𝑓-vector is𝑓(𝐴) + 3𝑓(𝑄) = (1, 16, 71, 98, 42). It is contractible butnot homeomorphic to a ball. (See Figure 7.)

The counterexample disproves several stronger conjec-tures, as discussed above: that constructible complexesare partitionable, Kalai’s conjecture about partitions in-dexed by algebraic shifting, and the Depth Conjecture.Yet the counterexample is not the end of the story, andmany questions remain. For instance, does the Partition-ability Conjecture hold in dimension two? Are simplicialballs partitionable? What about Garsia’s version of theconjecture for Cohen-Macaulay posets? What are the con-sequences for the theory of Stanley depth? What does theℎ-vector of a Cohen-Macaulay complex count?

The story of the Partitionability Conjecture has manyfacets. Shellability, partitionability, constructibility, andthe Cohen-Macaulay property come from different butoverlapping areas of mathematics: combinatorics, com-mutative algebra, topology, and discrete geometry. Thehierarchy of these structural properties turned out to bemore complicated thanwe had anticipated, just as Rudin’sand Ziegler’s examples demonstrate that even the sim-plest spaces canhave intricate combinatorics. Even thoughstatements like the Partitionability Conjecture can seemtoo beautiful to be false, we should remember to keepour minds open about the mathematical unknown—thereality might be quite different, with its own unexpectedbeauty.

References[DGKM16] Art M. Duval. Bennet Goeckner, Caroline J. Kli-

vans, and Jeremy L. Martin, A non-partitionableCohen-Macaulay simplicial complex, Adv. Math. 299(2016), 381–395. MR3519473

[PSFTY09] Mohammad R. Pournaki, Seyed A. Seyed Fakhari,Massoud Tousi, and Siamak Yassesmi, Whatis…Stanley depth? Notices Amer. Math. Soc. 56(9)(2009), 1106–1108. MR2568497

[Sta96] Richard P. Stanley, Combinatorics and Commuta-tive Algebra, second edition, volume 41 of Progressin Mathematics, Birkhäuser Boston, Inc., Boston, MA,1996. MR1453579

[Zie98] Günter M. Ziegler, Shelling polyhedral 3-balls and 4-polytopes, Discrete Comput. Geom. 19(2) (1998), 159–174. MR1600042

Photo CreditsFigure 6 is courtesy of Jennifer Wagner.The photo of Caroline Klivans, Art Duval, and Jeremy

Martin is courtesy of Matt Macauley.

ABOUT THE AUTHORS

Art Duval (center) is a professor at the University of Texas at El Paso and a con-tributing editor for the AMS blog “On Teaching and Learn-ing Mathematics.”

Caroline Klivans (left) is a lecturer in the Division of Applied Mathemat-ics and the Department of Computer Science at Brown University and an associate director of ICERM (Institute for Computational and Experimental Research in Math-ematics).

Jeremy Martin (right) is a professor at the University of Kansas and the vice chair of the Kansas Section of the Mathematical Association of America.

The three authors have a long-standing long-distance collaboration. Occasionally the three can be spotted in the same place at the same time.

Caroline Klivans, Art Duval, Jeremy Martin

122 Notices of the AMS Volume 64, Number 2