aa r X i v : . [ m a t h . C O ] A p r THE CUBICAL MATCHING COMPLEX REVISITED
DUŠKO JOJIĆ
Abstract.
Ehrenborg noted that all tilings of a bipartite planar graphare encoded by its cubical matching complex and claimed that this com-plex is collapsible. We point out to an oversight in his proof and explainwhy these complexes can be the union of collapsible complexes. Also, weprove that all links in these complexes are suspensions up to homotopy.Furthermore, we extend the definition of a cubical matching complexto planar graphs that are not necessarily bipartite, and show that thesecomplexes are either contractible or a disjoint union of contractible com-plexes.For a simple connected region that can be tiled with dominoes ( × and × ) and × squares, let f i denote the number of tilings withexactly i squares. We prove that f − f + f − f + · · · = 1 (establishedby Ehrenborg) is the only linear relation for the numbers f i . Introduction
Let G = ( V, E ) be a bipartite planar graph that allows a perfect matching.Assume that G is embedded in a plane. An elementary cycle of G is a cyclethat encircles a single region R different than outer region R ∗ . Throughoutthis paper, we identify an elementary cycle with the region it encircles aswell as with its set of vertices or edges.A tiling of G is a partition of the vertex set V into disjoint blocks of thefollowing two types:(1) an edge { x, y } of G ; or(2) an elementary cycle R (the set of vertices of R ).The set of all tilings of G form a cubical complex C ( G ) (called the cubicalmatching complex ) defined by Ehrenborg in [5]. Note that C ( G ) dependsnot only on G , but also on the choice of the embedding of that graph in theplane.A face F of C ( G ) has the form F = M F ∪ C F = ( M F , C F ) , where C F is acollection C F = { R , R , . . . , R t } of vertex-disjoint elementary cycles of G ,and M F is a perfect matching on G \ (cid:0) R ∪ R ∪ · · · ∪ R t (cid:1) . The dimensionof F is | C F | , and the vertices of C ( G ) are all perfect matchings of G .All tilings of G covered by F = ( M F , C F ) can be obtained by deletingan elementary cycle R from C F , and adding every other edge of R into M F (there are two possibilities to do this). Therefore, for two faces F =( M F , C F ) and F = ( M F , C F ) , we have that(1) (cid:0) F ⊂ F (cid:1) ⇐⇒ (cid:16) C F ⊂ C F and M F ⊃ M F (cid:17) . Key words and phrases. domino tilings, independence complexes, matching, cubicalcomplexes.
Let G ◦ denote the weak dual graph of a planar graph G . The vertices of G ◦ are all bounded regions of G , and two regions that share a common edgeare adjacent in G ◦ .The independence complex of a graph H is a simplicial complex I ( H ) whose faces are the independent subsets of vertices of H . Note that for anyface F = ( M F , C F ) of C ( G ) , the set C F contains independent vertices of G ◦ ,i.e., C F is a face of I ( G ◦ ) .At the first sight, the complex C ( G ) is related with the independencecomplex I ( G ◦ ) of its weak dual graph.A B C G G G C ( G ) C ( G ) C ( G ) A B C
A B CAC BA B C AC
Figure 1.
The three graphs with the same weak dual, butdifferent cubical matching complexes.However, Figure 1 shows the three graphs with the same weak dual butdifferent cubical matching complexes. The facets of the complexes on Fig-ure 1 are labeled by corresponding subsets of pairwise disjoint elementaryregions.
Example 1.
Let L n and C n denote the independence complexes of P n and C n (the path and cycle with n vertices) respectively. The homotopy typesof these complexes are determined by Kozlov in [9]: L n ≃ (cid:26) a point , if n = 3 k + 1 ; S ⌊ n − ⌋ , otherwise. C n ≃ (cid:26) S k − , if n = 3 k ± ; S k − ∨ S k − , if n = 3 k .We will use these complexes later, see Corollary 4 and Remark 7. Moredetails about combinatorial and topological properties of L n and C n (andabout the independence complexes in general), an interested reader can findin [6], [7] and [8].There are some cubical complexes that cannot be realized as subcomplexesof a d -cube C d = [0 , d , see Chapter of [4]. Proposition 2.
Let G be a bipartite planar graph that has a perfect match-ing. If G has d elementary regions, then its cubical matching complex C ( G ) can be embedded into C d . HE CUBICAL MATCHING COMPLEX REVISITED 3
Proof.
We use an idea from [10] to describe the coordinates of vertices of C ( G ) explicitly. Let R , R , . . . , R d be a fixed linear order of elementaryregions of G . We choose an arbitrary perfect matching M of G (a vertexof C ( G ) ) to be the origin = (0 , , . . . , in R d . For another vertex M of C ( G ) , we consider the symmetric difference M △ M . Note that M △ M is adisjoint union of cycles. For a given perfect matching M of G , we assign thevertex V M = ( x , . . . , x d ) of C d , where x i = (cid:26) , if R i is contained into an odd number of cycles of M △ M ; , otherwise.If M ′ and M ′′ are two perfect matchings of G such that M ′ △ M ′′ = R j (meaning that these two matchings differ just on an elementary region R j ),then their corresponding vertices V M ′ and V M ′′ of C d differ only at the j -thcoordinate.Therefore, the face F = ( M F , C F ) is embedded in C d as the convex hullof its | C ( F ) | vertices. (cid:3) The local structure of C ( G ) The star of a face F in a cubical complex C is the set of all faces of C thatcontain F star ( F ) = { F ′ ∈ C : F ⊂ F ′ } . The link of a vertex v in a cubical complex C is the simplicial complex link C ( v ) that can be realized in C as a “small sphere“ around the vertex v . More formally, the vertices of link C ( v ) are the edges of C containing v . A subset of vertices of link C ( v ) is a face of link C ( v ) if and only if thecorresponding edges belong to a same face of C .The link of a face F in a cubical complex C is defined in a similar way.The set of vertices of link C ( F ) is { F ′ ∈ C : F ⊂ F ′ and dim F ′ = 1 + dim F } , and a subset A of the set of vertices is a face of link C ( F ) if and only if allelements of A are contained in a same face of C .Ehrenborg investigated the links of the cubical complexes associated totilings of a region by dominos or lozenges.Here we describe the links in the cubical matching complex C ( G ) for anybipartite planar graph G . For a face F = ( M F , C F ) of C ( G ) , let R F denotethe set of all elementary regions of G for which every second edge is containedin M F . Further, let G F denote the subgraph of the weak dual graph G ◦ spanned with the regions from R F .From the definition of the link in a cubical complex and (1), we obtainthe next statement. Proposition 3.
For any face F = ( M F , C F ) of C ( G ) we have that link C ( F ) ∼ = I ( G F ) . The above proposition explains the appearance of complexes L n and C n as the links in cubical the matching complexes, see Theorem 3.3 and Section4 in [5]. DUŠKO JOJIĆ
Assume that all elementary regions of G are quadrilaterals. In that case,for any face F of C ( G ) , the degree of a vertex in G F is at most two. Therefore, G F is a union of paths and cycles. Corollary 4.
If all elementary regions of G are quadrilaterals, then link C ( F ) is a join of complexes L p and C q . Theorem 5.
Let G be a bipartite planar graph that has a perfect matching.For any face F = ( M F , C F ) of C ( G ) the graph G F is bipartite.Proof. Assume that G F contains an odd cycle R , R , . . . , R m +1 . Recallthat R i is an elementary region of G and the that every second edge of R i is contained in M F . Two neighborly regions R i and R i +1 have to share theodd number of edges, the first and the last of their common edges belongto M F . Therefore, for each region R i , there is an odd number of commonedges of R i and R i − that belong to M F . Obviously, the same holds for R i and R i +1 .So, we can conclude that there is an odd number of edges of R i that arebetween R i ∩ R i − and R i ∩ R i +1 (the first and the last one of these edgesare not in M F ). The union of all of these edges (for all regions R i ) is an oddcycle in G , which is a contradiction. (cid:3) Barmak proved in [1] (see also in [11]) that the independence complexesof bipartite graphs are suspensions, up to homotopy. This implies the nextresult.
Corollary 6.
All links in C ( G ) are homotopy equivalent to suspensions.Therefore, the link of any face in C ( G ) has at most two connected compo-nents. For any simplicial complex K there exists a bipartite graph G such thatthe independence complex of G is homotopy equivalent to the suspensionover K , see [1]. Skwarski proved in [12] (see also [1]) that there exists aplanar graph G whose independence complex is homotopy equivalent to aniterated suspension of K .We prove that the links of faces in cubical matching complexes are in-dependence complexes of bipartite planar graphs. What can be said abouthomotopy types of these complexes? Remark 7.
There is a natural question, posed by Ehrenborg in [5]:
Forwhat graphs G would the cubical matching complex C ( G ) be pure, shellable,non-pure shellable? The complexes L n are non-pure for n > , and the complexes C n are non-shellable for n > . Therefore, these complexes can be used to show that thecubical matching complex of a concrete graph is non-pure or non-shellable.3. Collapsibility and contractibility of cubical matchingcomplexes
The next theorem is the main result in [5].
Theorem 8 (Theorem 1.2 in [5]) . For a planar bipartite graph G that has aperfect matching, the cubical matching complex C ( G ) is collapsible. HE CUBICAL MATCHING COMPLEX REVISITED 5
The proof of the above statement is based on the next two results: ( i ) (Propp, Theorem 2 in [10]) The set of all perfect matchings of abipartite planar graph is a distributive lattice. ( ii ) (Kalai, see in [13], Solution to Exercise 3.47 c) The cubical complexof a meet-distributive lattice is collapsible.
Note however that Propp in his proof of ( i ) assumed the following two ad-ditional conditions for bipartite planar graph G : ( ∗ ) Graph G is connected, and ( ∗∗ ) Any edge of G is contained in some matching of G but not in others. Example 9.
The next figure shows a bipartite planar graph whose cubicalmatching complex is not collapsible. G C ( G ) Figure 2.
A bipartite planar graph G for which C ( G ) is not collapsible.Also, the Jockusch example (page 41 in [10], a bipartite planar graph with edges, but just of them can be used in a perfect matching), describe agraph G whose cubical matching complex is a disjoint union of four segments.The edges that do not appear in any perfect matching of a graph G (theforbidden edges) can be deleted. Also, if the edge xy is a forced edge ( xy appears in all perfect matching of G ), then we may consider the graph G −{ x, y } . e e e Figure 3.
If a new region can be included in a tiling of G − e ,then e is not forbidden. DUŠKO JOJIĆ
Remark 10.
Let e denote a forbidden edge in G and let G ′ = G − e . Thepossible new elementary region of G ′ , that appears after we delete e , cannot be included in a tiling of G ′ . Otherwise, we can find a perfect matchingof G that contains e , see Figure 3. In a similar way we conclude that thenew regions that appear after deleting a forced edge can not be included ina tiling of G ′ .Let G ′ denote the graph obtained from G after all deletions. Unfortu-nately, this new graph (after deleting all forced and forbidden edges) may benon-connected.If G ′ is connected, then the collapsibility of C ( G ′ ) follows from Ehrenborg’sproof. Also, if G ′ is non-connected, and all of its connected components areseparated (there is no component of G ′ that is contained in an elementaryregion of another component), then C ( G ′ ) is collapsible as a product of col-lapsible complexes.By using Remark 10, we can establish an obvious bijection between tilingsof G ′ and tilings of G (we just add all forced edges). Therefore, Theorem 8holds if G ′ is connected or if all of its connected components are separated.However, Theorem 8 fails if G ′ has two different connected components G and G such that G is contained in an elementary region R of G , seeExample 9. In that case we have that C ( G ′ ) = C ( G ) × ( C ( G ) \ { R } ) , and C ( G ′ ) is a union of collapsible complexes. Here C ( G ) \ { R } denotethe cubical complex obtained from C ( G ) by deleting all tilings (faces) thatcontain R as an elementary region. Figure 4.
Non-bipartite graphs and their cubical matching complexes.
HE CUBICAL MATCHING COMPLEX REVISITED 7
Now, we consider the cubical matching complex for all planar graphs thathave a perfect matching (not necessarily bipartite).
Definition 11.
Let G be a planar graph that allows a perfect matching.A tiling of G is a partition of the vertex set V into disjoint blocks of thefollowing two types: • an edge { x, y } of G ; or • the set of vertices { v , v , . . . , v m } of an even elementary cycle R .Let C ( G ) denote the set of all tilings of G . Note that C ( G ) is also a cubicalcomplex. Example 12. If G is a graph of a triangular prism (embedded in the planeso that the outer region is a triangle), then C ( G ) is a union of three -dimensional segments that share the same vertex, see the left side of Figure4. Each of segments of C ( G ) corresponds to a rectangle of prism. The linkof the common vertex of these segments is a -dimensional complex withthree points. Such situation is no possible for bipartite planar graphs, seeCorollary 6.The next theorem describe the homotopy type of the cubical matchingcomplex associated to a planar graph that allows a perfect matching. Theorem 13.
Let G be a planar graph that has a perfect matching. Thecubical complex C ( G ) is contractible or a disjoint union of contractible com-plexes. This is a weaker version (we prove contractibility instead collapsibility) ofcorrected Theorem 8, with a different proof.
Proof.
We use the induction on the number of edges of G . Let e = xy denote an edge that belongs to the outer region R ∗ . Let R = R ∗ denote theelementary region that contains e . If R is an odd region, then all tilings of G can be divided into two disjoint classes:(a) The tilings of G that do not use e . These tilings are just the tilingsof G \ e .(b) The tilings of G that contain e as an edge in a partial matchingcorrespond to the tilings of G \ { x, y } .In that case we obtain that C ( G ) = C ( G \ { x, y } ) ⊔ C ( G \ e ) is a disjoint unionof contractible complexes by inductive assumption.If R is an even elementary region, then some tilings of G may to contain R .Note that these tilings are not considered in (a) and (b). To describe thecorresponding faces of C ( G ) , we consider G \ R , the graph obtained from G by deleting all vertices from R .Let C e denote the subcomplex of C ( G \ e ) formed by all tilings that containevery second edge of R (but do not contain e , obviously). Further, let C x,y denote the subcomplex of C ( G \ { x, y } ) , defined by tilings that contain everysecond edge of R (these tilings have to contain e ). Note that the both ofcomplexes C e and C x,y are isomorphic to C ( G \ R ) . In that case we obtain(2) C ( G ) = C ( G \ { x, y } ) ∪ C ( G \ e ) ∪ P rism ( C ( G \ R )) . DUŠKO JOJIĆ
Further, we have that C ( G \ e ) ∩ P rism ( C ( G \ R )) = C e and C ( G \ { x, y } ) ∩ P rism ( C ( G \ R )) = C x,y . The complexes on the right-hand side of (2) are disjoint unions of con-tractible complexes by the inductive hypothesis. Assume that C ( G \ { x, y } ) = A ⊔ A ⊔ · · · ⊔ A s and C x,y = B ⊔ B ⊔ · · · ⊔ B t , where A i and B j denote the contractible components of corresponding com-plexes. Obviously, each complex B j is contained in some A i . Now, we needthe following lemma. Lemma 14.
Each of connected component of C ( G \ { x, y } ) contains at mostone component of C x,y .Proof of Lemma: Assume that a component of C ( G \ { x, y } ) contains twocomponents of C x,y . In that case, there are two vertices of C x,y (perfectmatchings of G that contain xy ) that are in different components of C x,y ,but in the same component of C ( G \ { x, y } ) . Assume that M ′ and M ′′ aretwo such vertices, chosen so that the distance between them in C ( G \ { x, y } ) is minimal. Let(3) M ′ = M R M . . . M i R i M i +1 . . . M n R n M n +1 = M ′′ denote the shortest path from M ′ to M ′′ in C ( G \ { x, y } ) . The perfectmatching M i +1 is obtained from M i by removing the edges of M i containedin an elementary region R i , and by inserting the complementary edges. Inother words, we have that M i +1 = M i △ R i , for an elementary region R i contained in R F i ∩ R F i +1 .Note that R must be adjacent (share the common edge) with R . Oth-erwise, both of vertices M and M belong to the same component of C x,y ,and we obtain a contradiction with the assumption that the path describedin (3) is minimal.In a similar way, we obtain that for any i = 1 , , . . . , n , the region R i must be adjacent with at least one of regions R, R , R , . . . , R i − . If not, wehave that the perfect matching M = M △ R i belongs to C x,y , and M and M ′ are contained in the same component of C x,y . In that case we obtain acontradiction, because the path M = M R M . . . M i − R i − M i +1 R i +1 . . . M n R n M n +1 = M ′′ is shorter than (3). Here we let that M j +1 = M j △ R j .Let e ′ denote a common edge of regions R and R that is contained in M ′ .Note that e ′ is not contained in M . However, this edge is again containedin M ′′ , and we conclude that the region R has to reappear again in (3).Let R i = R denote the first appearance of R in (3) after the first step.There are the following three possible situations that enable the reappearanceof R : ( a ) All regions R k (for k between and i ) are disjoint with R .In that case, we can omit the steps in (3) labelled by R and R i ,and obtain a shorter path between M ′ and M ′′ . HE CUBICAL MATCHING COMPLEX REVISITED 9 ( b ) Any region that shares at least one edge with R appears an oddnumber of times between R and R i .This is impossible, because R (that share an edge with R ) can notappear in (3). ( c ) There is t < i such that R t = ¯ R shares an edge with R , but thefragment of the sequence (3) between R and R i does not containall region that shares an edge with R .Then the same region ¯ R has to appear again as R s , for some s suchthat t < s < i . Again, if all regions R j are disjoint with ¯ R (for j = t + 1 , . . . , s − ), we can omit R t and R s , and obtain a contradiction.If not, there exist indices t ′ and s ′ such that t < t ′ < s ′ < s and R ′ t = R ′ s . We continue in the same way, and from the finiteness ofthe path, obtain a shorter path than (3). (cid:3) Continue of Proof:
We built C ( G ) by starting with C ( G \ e ) , that is a unionof contractible complexes by assumption. Then we glue the components of P rism ( C ( G \ R )) one by one.After that, we glue all components of C ( G \ { x, y } ) . At each step we aregluing two contractible complexes along a contractible subcomplex, or wejust add a new contractible complex, disjoint with previously added compo-nents. From the Gluing Lemma (see Lemma 10.3 in [3]) we obtain that C ( G ) is contractible, or a disjoint union of contractible complexes. (cid:3) Remark 15.
For a connected bipartite planar graph G that satisfy the con-dition ( ∗∗ ) , the cubical matching complex C ( G ) is collapsible, see Theorem8. The planar graph on the right side on Figure 4 satisfies the condition ( ∗∗ ) , but the corresponding cubical complex is not collapsible, it is a unionof three disjoint edges. So, there is a natural question: Is there a property of G that provides the collapsibility of its cubical com-plex C ( G ) ? Obviously, if all complexes that appear on the right-hand sideof (2) are nonempty and contractible, then C ( G ) is contractible.4. The f -vector of domino tilings The concept of tilings of a bipartite planar graph generalizes the notionof domino tilings. Let R be a simple connected region, compound of unitsquares in the plane, that can be tiled with domino tiles × and × .The set of all tilings of R by domino tiles and × squares defines a cubicalcomplex, denoted by C ( R ) . If we consider R as a planar graph (all of itselementary regions are unit squares), and if G denotes the weak dual graphof R (the unit squares of R are vertices of G ), then C ( R ) is isomorphic tothe cubical matching complex C ( G ) , see Section 3 in [5] for details. Notethat the number of i -dimensional faces of C ( G ) counts the number of tilingsof R with exactly i squares × .Ehrenborg used collapsibility of C ( G ) to conclude (see Corollary 3.1. in[5]) that the entries of f -vector of f ( C ( G )) = ( f , f , . . . , f d ) satisfy(4) f − f + f − · · · + ( − d f d = 1 . If G is the weak dual graph of a region R that admits a domino tiling,then all complexes that appear on the right-hand side of the relation (2) arecontractible by induction, and therefore C ( G ) is contractible, see Remark15. So, we obtain that the relation (4) is true in any case, disregardingpossible problems with Theorem 8. In this Section we will prove that (4)is the only linear relation for f -vectors of cubical complexes of domino tilings.For all n ∈ N , we let G n denote the following graph n . This graph (also known as the ladder graph) has n + 2 vertices, n + 1 edgesand n elementary regions (squares). For i = 1 , , . . . , n , let G n,i denote thegraph obtained by adding one unit square below the i -th square of G n . Now,we describe some recursive relations for f -vectors of C ( G n ) and C ( G n,i ) . Proposition 16.
The entries of f -vectors of C ( G n ) and C ( G n,i ) satisfy thefollowing recurrences: (5) f i ( C ( G n +2 )) = f i ( C ( G n +1 )) + f i ( C ( G n )) + f i − ( C ( G n )) , (6) f i ( C ( G n +2 ,i )) = f i ( C ( G n +1 ,i )) + f i ( C ( G n,i )) + f i − ( C ( G n,i )) , (7) f i ( C ( G n +2 ,i )) = f i ( C ( G n +1 ,i − )) + f i ( C ( G n,i − )) + f i − ( C ( G n,i − )) . Proof.
All formulas follow from relation (2), see the proof of Theorem 13.To obtain the formula (5), we apply (2) on G n +2 . The rightmost verticaledge and the rightmost unit square in G n +2 act as e and R in (2). = ⊔ ⊔ = ⊔ ⊔ = ⊔ ⊔ (5)(6)(7) Figure 5.
The “geometric proof“ of recursive relations for f ( C ( G n )) and f ( C ( G n,i )) .In the same way we can prove the remaining two relations. For eachrelation, we choose an adequate elementary region R , a corresponding edge e of R , and use relation (2), see Figure 5. (cid:3) The f -vector ( f , f , f , . . . , f ⌈ n ⌉ ) of C ( G n ) can be encoded by the polynomial F n : F n = F C ( G n ) ( x ) = f + f x + f x + · · · + f ⌈ n ⌉ x ⌈ n ⌉ . Similarly, we define the polynomials F n,i to encode the f -vector of C ( G n,i ) .Directly from (5) and (6) we obtain that F n +2 ( x ) = F n +1 ( x ) + ( x + 1) F n ( x ) , F n +2 ,i ( x ) = F n +1 ,i ( x ) + ( x + 1) F n,i ( x ) . HE CUBICAL MATCHING COMPLEX REVISITED 11
Now, we define new polynomials P n and P n,i by P n = P n ( x ) = F n ( x − , P n,i = P n,i ( x ) = F n,i ( x − . This is a variant of h -polynomial associated to corresponding cubical com-plexes.From Proposition 16 it follows that the polynomials P n and P n,i satisfy thefollowing recurrences(8) P n +2 ( x ) = P n +1 ( x ) + xP n ( x ) , (9) P n +2 ,i ( x ) = P n +1 ,i ( x ) + xP n,i ( x ) , (10) P n +2 ,i ( x ) = P n +1 ,i − ( x ) + xP n,i − ( x ) . Remark 17.
We can use (8) to obtain the polynomials P n explicitly P d − = (cid:18) dd (cid:19) x d + · · · + (cid:18) d + kd − k (cid:19) x k + · · · + (cid:18) d − (cid:19) x + (cid:18) d (cid:19) , and P d = (cid:18) d + 1 d (cid:19) x d + · · · + (cid:18) d + k + 1 d − k (cid:19) x k + · · · + (cid:18) d (cid:19) x + (cid:18) d + 10 (cid:19) . Note that the polynomials P n are related with Fibonacci polynomials, seeSection 9.4 in [2] for the definition and a combinatorial interpretation ofcoefficients. The coefficient of these polynomials are positive integers andthe sum of coefficients of P n is a Fibonacci number. Note that this is justthe number of vertices in C ( G n ) .Assume that we embedded C ( G n ) into n -cube as in Proposition 2, so thatthe perfect matching M = of G n is the vertex in the origin.Now, the coefficient of x k in P n counts the number of vertices of C ( G n ) forwhich the sum of coordinates is k , i.e., it is the number of vertices of C ( G n ) whose distance from M is k .Also, following [2], we can recognize the coefficient of x k in P n as thenumber of k -element subsets of [ n ] that do not contain two consecutive inte-gers. Similarly, we can interpret the coefficient of x k in P n,i as the numberof k -element subsets of the multiset M = { , , . . . , i − , i, i, i + 1 , . . . , n } that do not contain two consecutive integers. Note that the multiplicity of i in M is two, and all other elements have the multiplicity one. Definition 18.
Let P d denote the vector space of all polynomials of degreeat most d . We define the linear map A d : P d → P d +1 recursively by(11) A d ( x k ) = xA d − ( x k − ) for all k > , (12) A (1) = 1 + 2 x and A d (1) = P d +1 − A d ( P d − − . Lemma 19.
For any non-negative integer d , we have that A d ( P d − ) = P d +1 , A d ( P d ) = P d +2 and A d +1 ( P d ) = P d +2 . Proof.
From (12) it follows that A d ( P d − ) = P d +1 . For the proof of thesecond formula we use (8), (11) and induction A d ( P d ) = A d ( P d − + xP d − ) = P d +1 + xA d − ( P d − ) = P d +1 + xP d = P d +2 . The last formula in this lemma follows from (8) and earlier proved formulas A d +1 ( P d ) = A d +1 ( P d +1 − xP d − ) = P d +3 − xA d ( P d − ) = P d +3 − xP d +1 = P d +2 . (cid:3) Lemma 20.
For all integers i and d such that ≤ i ≤ ⌊ d ⌋ , the followingholds: A d ( P d − ,i ) = P d +1 ,i and A d ( P d,i ) = P d +2 ,i . Proof.
For i = 1 and i = 2 we apply relation (2) in a similar way as in theproof of Proposition 16. We just delete the only square in the second row of G n, and G n, , and obtain that P d − , = P d − + xP d − , P d − , = P d − + xP d − . By using Lemma 19, we obtain that A d ( P d − , ) = A d ( P d − + xP d − ) = P d +1 + xP d − = P d +1 , , and A d ( P d − , ) = A d ( P d − + xP d − ) = P d +1 + xA d − ( P d − ) == P d +1 + xP d − = P d +1 , . In a similar way, we can prove that A d ( P d, ) = P d +2 , , A d ( P d, ) = P d +2 , . Assume that the statement of this lemma is true for P d − ,j and P d,j when j < i + 1 . Now, we use (10) and induction to calculate A d ( P d,i +1 ) = A d ( P d − ,i + xP d − ,i − ) = A d ( P d − ,i ) + xA d − ( P d − ,i − ) == P d +1 ,i + xP d,i − = P d +2 ,i +1 . From (9) we obtain that A d ( P d − ,i +1 ) = A d ( P d,i +1 − xP d − ,i +1 ) = A d ( P d,i +1 ) − xA d − ( P d − ,i +1 ) == P d +2 ,i +1 − xP d,i +1 = P d +1 ,i +1 . (cid:3) From Definition 18 and Remark 17 we can obtain the concrete formula forthe linear map A d . Proposition 21.
For all d, k ∈ N such that d ≥ k ≥ , we have that: A d ( x k ) = x k (cid:16) x − x + 2 x − x + 14 x − · · · + ( − d − k C d − k x d − k +1 (cid:17) . Here C m denotes the m -th Catalan number. HE CUBICAL MATCHING COMPLEX REVISITED 13
Proof.
From (11) it is enough to prove that(13) A d (1) = 1 + 2 x − x + 2 x − x + · · · + ( − d C d x d +1 . For all integers n and k such that n ≥ k ≥ (by using the induction andthe Pascal’s Identity), we can obtain the next relation(14) (cid:18) nk (cid:19) = k X i =0 ( − i (cid:18) n + 1 + ik − i (cid:19) C i . Now, we assume that (13) is true for all positive integers less than d , andcalculate A d (1) by definition: A d (1) = P d +1 − A d ( P d − −
1) == d +1 X i =0 (cid:18) d + 2 − ii (cid:19) x i − d X i =1 (cid:18) d − ii (cid:19) x i A d − i (1) . The coefficients of , x and x in A d (1) are respectively: (cid:18) d + 20 (cid:19) = 1 , (cid:18) d + 11 (cid:19) − (cid:18) d − (cid:19) = 2 , (cid:18) d (cid:19) − (cid:18) d − (cid:19) − (cid:18) d − (cid:19) = − . For k > the coefficient of x k +1 in the polynomial A d (1) is (cid:18) d + 1 − kk + 1 (cid:19) − (cid:18) d − k − k + 1 (cid:19) − (cid:18) d − kk (cid:19) − k − X i =1 ( − i (cid:18) d − k + ik − i (cid:19) C i . From (14) we obtain that the coefficient of x k +1 in A d (1) is ( − k C k . (cid:3) Corollary 22.
For any positive integer d the linear map A d is injective. Now, we consider all simple connected regions for which the degree of theassociated polynomial P R ( x ) = F R ( x − is equal to d . Let F d denote theaffine subspace of P d spanned by these polynomials. Lemma 23.
The polynomial P d +1 ,d is not contained in A d ( F d ) .Proof. From (10) and (9) we have that P d +1 ,d − P d +1 ,d − = ( P d,d − + xP d − ,d − ) − ( P d,d − + xP d − ,d − ) = − x ( P d − ,d − − P d − ,d − ) = ( − d +1 ( x d +1 + x d ) . We know that P d +1 ,d − = A d ( P d − ,d − ) . If there exists a polynomial p ∈F d such that A d ( p ) = P d +1 ,d then we obtain x d +1 + x d = ± A d ( p − P d − ,d − ) , which is impossible from Proposition 21. (cid:3) Theorem 24.
The polynomials P d − , P d , P d − , , . . . , P d − ,d − are affinelyindependent in F d . Proof.
We use induction on the degree. Assume that d polynomials P d − , P d − , P d − , , . . . , P d − ,d − are affinely independent in F d − . From Lem-mas 19 and 20 and Corollary 22, we conclude that P d − , P d , P d − , , . . . , P d − ,d − are affinely independent. These polynomials span a ( d − -dimensional affine subspace of F d . From Lemma 23 follows that P d − ,d − is not contained in A d − ( F d − ) . (cid:3) Corollary 25.
The Euler-Poincare relation (4) is the only linear relationfor the f -vectors of tilings. This answer the question of Ehrenborg question about numerical relationsbetween the numbers of different types of tilings, see [5].
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