simplicial matrix-tree theorems - utep mathematics · 2007-07-15 · simplicial matrix-tree...

SIMPLICIAL MATRIX-TREE THEOREMS

ART M. DUVAL, CAROLINE KLIVANS, AND JEREMY L. MARTIN

Abstract. We generalize the definition of a spanning tree from graphs to arbitrary simplicial

complexes ∆, following G. Kalai. We show that the simplicial spanning trees of a complex can be

counted, weighted by the squares of the orders of their top-dimensional integral homology groups,using its Laplacian matrix. A more refined count of trees can be obtained by assigning a variable

to each vertex of ∆ and replacing the Laplacian with a monomial matrix. These results generalize

the classical matrix-tree theorem, and its weighted analogue, from graphs to complexes. When ∆is a shifted complex, we give a complete description of the eigenvalues of the weighted Laplacian in

terms of the shifted order on its faces, generalizing previously known results for threshold graphs

and for unweighted Laplacian eigenvalues.

Contents

1. Introduction 22. Notation and definitions 22.1. Simplicial complexes 22.2. Multisets 22.3. Combinatorial Laplacians 23. Simplicial spanning trees 34. Simplicial analogues of the matrix-tree theorem 55. Weighted enumeration of simplicial spanning trees 86. Shifted complexes and (the start of) counting their spanning trees 97. The vertex weighting 108. Vertex-weighting, cones, and near-cones 128.1. Boundary and coboundary operators of cones 128.2. Eigenvectors of cones 148.3. The vertex weighting and near-cones 169. z-polynomials and critical pairs 179.1. z-polynomials 179.2. Critical pairs 199.3. The theorem, and its consequences 219.4. An example: the bipyramid 2310. Return to spanning trees of shifted complexes 2511. Corollaries 2711.1. Fine enumerator from shifted to threshold 2711.2. Fine enumerator from threshold to Ferrers 28References 29

Date: July 15, 2007.

1

2 ART M. DUVAL, CAROLINE KLIVANS, AND JEREMY L. MARTIN

1. Introduction

2. Notation and definitions

2.1. Simplicial complexes. Let V be a finite set. A simplicial complex on V is a family ∆ ofsubsets of V such that

(1) ∅ ∈ ∆;(2) If F ∈ ∆ and G ⊂ F , then G ∈ ∆.

The elements of V are called vertices of ∆, and the faces that are maximal under inclusion are calledfacets. The dimension of a face F is dimF = |F | − 1, and the dimension of ∆ is the maximumdimension of a face (or facet). The abbreviation ∆d indicates that dim ∆ = d. We say that ∆ ispure if all facets have the same dimension; in this case, a ridge is a face of dimension dim ∆− 1.

Let ∆i be the set of i-dimensional faces of ∆, and let fi(∆) = |∆i|. The i-dimensional skeletonof ∆ is the subcomplex

∆(i) =⋃

−1≤j≤i

∆j .

The pure i-dimensional skeleton of ∆ is the subcomplex ∆[i] generated by the i-dimensional faces,that is,

∆[i] = F ∈ ∆: F ⊆ G, for some G ∈ ∆i.We assume that the reader is familiar with simplicial homology (see, e.g., [13, §2.1]). Let ∆d be

a simplicial complex and −1 ≤ i ≤ d. Let R be a ring (if unspecified, assumed to be Z), and letCi(∆) be the ith simplicial chain group, i.e., the free R-module with basis [σ] : σ ∈ ∆i. We definethe simplicial boundary and coboundary maps respectively by

∂∆,i : Ci(∆)→ Ci−1(∆),

∂∗∆,i : Ci−1(∆)→ Ci(∆)

(where we have identified cochains with chains via the natural inner product). We will abbreviate thesubscripts in the notation for boundaries and coboundaries whenever no ambiguity can arise. We willoften regard ∂i (resp. ∂∗i ) as a matrix whose columns and rows (resp. rows and columns) are indexedby ∆i and ∆i−1 respectively. The ith (reduced) homology group of ∆ is Hi(∆) = ker(∂i)/ im(∂i+1),and the ith (reduced) Betti number βi(∆) is the rank of the largest free R-module summand ofHi(∆).

2.2. Multisets. We will often work with multisets (of eigenvalues or of vertices), in which eachelement may occur with positive integer multiplicity. For brevity, we drop curly braces and commaswhen working with multisets of integers: for instance, 5553 denotes the multiset in which 5 occurswith multiplicity three and 3 occurs with multiplicity one. The cardinality of a multiset is the sumof the multiplicities of its elements; thus |5553| = 4. We write a = b to mean that the multisetsa and b differ only in their respective multiplicities of zero; for instance, 5553 = 55530 = 555300.Of course, = is an equivalence relation.) The union operation ∪ on multisets is understood to addmultiplicities: for instance, 5553 ∪ 5332 = 55553332.

2.3. Combinatorial Laplacians. We adopt the notation of [9] for the Laplacian operators (ormatrices) of a simplicial complex. We summarize the notation and mention some fundamentalidentities here.

For −1 ≤ i ≤ dim ∆, define linear operators Lud∆,i, L

du∆,i, L

tot∆,i on the vector space Ci(∆) by

Lud∆,i = ∂i+1∂

∗i+1 (the up-down Laplacian),

Ldu∆,i = ∂∗i ∂i (the down-up Laplacian),

Ltot∆,i = Lud

∆,i + Ldu∆,i (the total Laplacian).

SIMPLICIAL MATRIX-TREE THEOREMS 3

Let stoti (∆) be the spectrum of Ltot

∆,i; that is, the multiset of its eigenvalues (including zero).Similarly, let sud

i (∆) and sdui (∆) be the spectra of Lud

∆,i and Ldu∆,i respectively. Since each Laplacian

operator is represented by a symmetric matrix, it is diagonalizable, so

|stoti (∆)| = |sud

i (∆)| = |sdui (∆)| = fi(∆).

The various Laplacian spectra are related by the identities

sudi (∆) = sdu

i+1(∆),

stoti (∆) = sud

i (∆) ∪ sdui (∆)

[9, eqn. (3.6)]. Therefore, each of the three families of multisets

stoti (∆): − 1 ≤ i ≤ dim ∆, sud

i (∆): − 1 ≤ i ≤ dim ∆, sdui (∆): − 1 ≤ i ≤ dim ∆

determines the other two, and we will feel free to work with whichever one is most convenient incontext.

3. Simplicial spanning trees

Definition 3.1. Let ∆ be a d-dimensional simplicial complex, and let k ≤ d. A k-dimensionalsimplicial spanning tree (for short, SST or k-SST) of ∆ is a k-dimensional subcomplex Υ ⊆ ∆ suchthat Υ(k−1) = ∆(k−1) and

Hk(Υ) = 0, (1a)

|Hk−1(Υ)| <∞, and (1b)

fk(Υ) = fk(∆)− βk(∆) + βk−1(∆). (1c)

We write Tk(∆) for the set of all k-SSTs of ∆, omitting the subscript if k = d. Note thatTk(∆) = T (∆(j)) for all j ≥ k.

An 0-dimensional spanning tree is just a vertex of ∆. If ∆ is a 1-dimensional simplicial complexon n vertices—that is, a graph—then the definition of 1-SST coincides with the usual definition ofa spanning tree of a graph. Namely, a subgraph of ∆ which is connected, acyclic, and has n − 1edges. Next, we give a few examples in higher dimensions.

Example 3.2. If ∆ is a simplicial sphere (for instance, the boundary of a simplicial polytope),then deleting any facet of ∆ produces a simplicial spanning tree. Hence a k-dimensional simplicialsphere has fk(∆) SSTs.

Example 3.3. In dimension > 1, spanning trees need not be acyclic. For example, let ∆ be atriangulation of the real projective plane, so that dim ∆ = 2 and H2(∆) ∼= Z/2Z; then ∆ is a 2-SSTof itself.

Example 3.4. Consider the two-dimensional complex with facets 123, 124, 125, 134, 135, 234, 235,the “bipyramid with equator”. There are two ways to obtain a SST of this complex; remove the“equator” facet 123 and any one boundary facet or remove one boundary facet from each of the twopyramids of the bipyramid. Thus the number of SSTs is 15: six of the first type and nine of thesecond.

The next fact generalizes the fundamental property of spanning trees of graphs that any two ofthe three conditions–connected, acyclic, has n− 1 edges–implies the third.

Proposition 3.5. Let Υ ⊂ ∆ be a subcomplex with Υ(k−1) = ∆(k−1). Then any two of the conditions(1a), (1b), (1c) imply the third.


Proof. Note that

f`(Υ) = f`(∆) for ` ≤ k − 1, β`(Υ) = β`(∆) for ` ≤ k − 2. (2)

We calculate the Euler characteristic χ(Υ) in two ways, as the alternating sum of the f -numbersand of the Betti numbers. First,

χ(Υ) =k∑i=0

(−1)ifi(Υ)

= (−1)kfk(Υ) +k−1∑i=0

(−1)ifi(∆)

= (−1)kfk(Υ) + χ(∆)− (−1)kfk(∆). (3)

On the other hand

χ(Υ) =k∑i=0

(−1)iβi(Υ)

= (−1)k(βk(Υ)− βk−1(Υ)) +k−2∑i=0

(−1)iβi(∆)

= (−1)k(βk(Υ)− βk−1(Υ)) + χ(∆)− (−1)k(βk(∆)− βk−1(∆)). (4)

Equating (3) and (4) gives

fk(Υ)− fk(∆) = βk(Υ)− βk−1(Υ)− βk(∆) + βk−1(∆)

or equivalently (fk(Υ)− fk(∆) + βk(∆)− βk−1(∆)

)− βk(Υ) + βk−1(Υ) = 0.

Since (1a) says that βk(Υ) = 0 (note that Hk(Υ) must be free abelian) and (1b) says that βk−1(Υ) =0, the conclusion follows.

Definition 3.6. A simplicial complex ∆ is called metaconnected if βj(∆) = 0 for all j < dim ∆.

Equivalently, a d-dimensional complex is metaconnected if it has the homology type of a wedge ofzero or more d-dimensional spheres. In particular, any Cohen-Macaulay complex is metaconnected.The converse is very far from true, because, for instance, a metaconnected complex need not even bepure. For our purposes, metaconnected complexes are the “right” simplicial analogues of connectedgraphs for the reason stated in the next proposition.

Proposition 3.7. A simplicial complex ∆ has a simplicial spanning tree if and only if it is meta-connected.

Proof. Let d = dim ∆. Suppose that βk(∆) = 0 for all k < d. Then for each k < d, we can constructa spanning k-tree Υ by taking the entire k-skeleton of ∆ and removing facets until we get a complexwhose k-dimensional homology vanishes. On the other hand, suppose that k < d and that Υ is asimplicial spanning (k + 1)-tree of ∆. Consider the following diagram:

Ck+1(∆)∂∆,k−−−→ Ck(∆)

∂∆,k−1−−−−→ Ck−1(∆)⋃‖ ‖

Ck+1(Υ)∂Υ,k−−−→ Ck(Υ)

∂Υ,k−1−−−−→ Ck−1(Υ)

Then ker ∂∆,k−1 = ker ∂Υ,k−1 and im ∂∆,k ⊇ im ∂Υ,k, so there is a surjection Hk(Υ) → Hk(∆).The domain is finite by definition of Υ, so the target is also finite.


Metaconnectedness is a fairly mild condition on simplicial complexes. Any Z-acyclic complex ismetaconnected (and is its own unique spanning tree), as is any Cohen-Macaulay complex. In par-ticular, shifted and matroid complexes, which we will study in more detail later, are metaconnected.

4. Simplicial analogues of the matrix-tree theorem

Throughout this section, let ∆d be a metaconnected simplicial complex. For k ≤ d, define

πk = πk(∆) =∏

06=λ∈sudk−1(∆)

λ, hk = hk(∆) =∑

Υ∈Tk(∆)

|Hk−1(Υ)|2.

We are interested in the relationships between these two families of invariants. When d = 1, therelationship is given by the classical matrix-tree theorem; see, e.g., [17, 20].

Theorem (Matrix-Tree Theorem I—product of nonzero eigenvalues). Let ∆ be a graph(i.e., a 1-dimensional simplicial complex) on n vertices. Then

h1 =π1

n.

The matrix-tree theorem is often reformulated in terms of the reduced Laplacian. Let ∂ = ∂∆,1

be the simplicial boundary matrix of ∆ (equivalently, the signed vertex-edge incidence matrix).Deleting the row of ∂ corresponding to a vertex i produces the reduced boundary matrix ∂i. Thereduced Laplacian is then Li = ∂i∂

∗i ; this is an (n− 1)× (n− 1) symmetric matrix.

Theorem (Matrix-Tree Theorem II—determinant of reduced Laplacian). With the nota-tion above, we have for all i

h1 = detLi.

The goal of this section is to generalize both versions of the matrix-tree theorem to arbitrarymetaconnected complexes ∆d. Our results extend the work of Kalai [15] and Adin [1] on simplicialgeneralizations of complete graphs and complete bipartite graphs respectively. Indeed, Kalai’s andAdin’s arguments can be generalized to all metaconnected complexes, and we proceed by mimickingtheir approach.

Throughout the rest of the section, abbreviate βi = βi(∆), fi = fi(∆), and ∂ = ∂∆,d. Let T bea set of facets of ∆ of cardinality fd − βd + βd−1 = fd − βd, and let S be a set of ridges such that|S| = |T |. Define

∆T = T ∪∆(d−1), S = ∆(d−1) \ S, ∆S = S ∪∆(d−2), ∂ = ∂∆,d.

Also, let ∂S,T be the square submatrix of ∂ with rows indexed by S and columns indexed by T .

Proposition 4.1. The matrix ∂S,T is nonsingular if and only if ∆T ∈ Td(∆) and ∆S ∈ Td−1(∆).

Proof. We may regard ∂S,T as the top boundary map of the d-dimensional relative complex Γ =(∆T ,∆S). So ∂S,T is nonsingular if and only if Hd(Γ) = 0. Consider the long exact sequence

0→ Hd(∆S)→ Hd(∆T )→ Hd(Γ)→ Hd−1(∆S)→ Hd−1(∆T )→ Hd−1(Γ)→ · · · (5)

If Hd(Γ) 6= 0, then Hd(∆T ) and Hd−1(∆S) cannot both be zero. This proves the “only if” direction.If Hd(Γ) = 0, then Hd(∆S) = 0 (since dim ∆S = d−1), so (5) implies Hd(∆T ) = 0. Therefore ∆T

is a d-tree, because it has the right number of facets. Hence Hd−1(∆T ) is finite. Then (5) impliesthat Hd−1(∆S) is finite. In fact, it is zero because the top homology group of any complex must betorsion-free. Meanwhile, ∆S has the right number of facets to be a simplicial spanning (d− 1)-treeof ∆, proving the “if” direction.

Proposition 4.2. If ∂S,T is nonsingular, then

|det ∂S,T | =|Hd−1(∆T )| · |Hd−2(∆S)|

|Hd−2(∆T )|=|Hd−1(∆T )| · |Hd−2(∆S)|

|Hd−2(∆)|.


Proof. As before, we interpret ∂S,T as the boundary map of the relative complex Γ = (∆T ,∆S). So∂S,T is a map from Z|T | to Z|T |, and Z|T |/∂S,T (Z|T |) is a finite abelian group of order |det ∂S,T |. Onthe other hand, since Γ has no faces of dimension ≤ d− 2, its lower boundary maps are all zero, so|det ∂S,T | = |Hd−1(Γ)|. Since Hd−2(∆T ) is finite, the desired result now follows from the piece

0→ Hd−1(∆T )→ Hd−1(Γ)→ Hd−2(∆S)→ Hd−2(∆T )→ 0 (6)

of the long exact sequence (5).

Theorem 4.3 (Simplicial Matrix-Tree Theorem I—product of non-zero eigenvalues). Let∆ be a metaconnected complex of dimension d. Then

πd(∆) =hd(∆)hd−1(∆)|Hd−2(∆)|2

.

Proof. Let L = Lud∆,d−1. This is a square matrix whose size is fd−1 and whose rank is fd − βd =

fd − βd + βd−1 (by metaconnectedness). Let χ(L; y) = det(yI −L) be its characteristic polynomial,so that πd(∆), the product of the nonzero eigenvalues of L, is given (up to sign) by the coefficientof yfd−1−fd+βd in χ(L; y). Equivalently,

πd =∑

S⊂∆d−1|S|=rankL

detLS =∑

S⊂∆d−1|S|=fd−βd

detLS , (7)

where LS denotes the square submatrix of L with columns and rows indexed by S. By the Binet-Cauchy formula, we have

detLS =∑T⊂∆d

|T |=|S|

(det ∂∗T,S)(det ∂S,T ) =∑T⊂∆d

|T |=|S|

(det ∂S,T )2. (8)

Combining (7) and (8), applying Proposition 4.1, and interchanging the sums, we obtain

πd =∑

T∈Td(∆)

∑S∈Td−1(∆)

(det ∂S,T )2

and now applying Proposition 4.2 yields

πd =∑

T∈Td(∆)

∑S∈Td−1(∆)

(|Hd−1(∆T )| · |Hd−2(∆S)|

|Hd−2(∆)|

)2

=

∑T∈Td(∆)

|Hd−1(∆T )|2 ∑

S∈Td−1(∆)

|Hd−2(∆S)|

|Hd−2(∆)|2

as desired.

We next seek a simplicial analogue of the second version of the matrix-tree theorem. First, wecheck that when we delete the rows of ∂ corresponding to a simplicial spanning (d − 1)-tree, theresulting reduced Laplacian has the correct size, namely, that of a spanning d-tree.

Lemma 4.4. Let U ∈ Td−1(∆) and S = ∆d−1\U . Then |S| = fd(∆)− βd(∆), the number of facetsof a spanning tree of ∆.


Proof. Let Γ = ∆(d−1). By Proposition 3.5 and the observation (2), |U | = fd−1(Γ) − βd−1(Γ) +βd−2(Γ) = fd−1(∆)− βd−1(Γ), so |S| = βd−1(Γ). . The Euler characteristics of ∆ and Γ are

χ(∆) =d∑i=0

(−1)ifi(∆) =d∑i=0

(−1)iβi(∆),

χ(Γ) =d−1∑i=0

(−1)ifi(Γ) =d−1∑i=0

(−1)iβi(Γ).

By (2), we see that

χ(∆)− χ(Γ) = (−1)dfd(∆) = (−1)dβd(∆) + (−1)d−1βd−1(∆)− (−1)d−1βd−1(Γ)

from which we obtain fd(∆) = βd(∆) − βd−1(∆) + βd−1(Γ). Since ∆ is metaconnected, we haveβd−1(∆) = 0, so |S| = βd−1(Γ) = fd(∆)− βd(∆) as desired.

Theorem 4.5 (Simplicial Matrix Tree Theorem II—determinant of reduced Laplacian).Let U ∈ Td−1(∆) and S = ∆d−1\U . Let ∂S be the reduced boundary matrix obtained from ∂ bydeleting the rows corresponding to ridges in U , and let LS = ∂S∂

∗S. Then

hd(∆) =|Hd−2(∆)|2

|Hd−2(∆U )|2detLS .

Proof. The Binet-Cauchy formula gives

detLS = det(∂S∂∗S) =∑

T : |T |=|S|

(det ∂S,T )(det ∂∗S,T )

=∑

T : |T |=|S|

(det ∂S,T )2.

By Lemma 4.4 and Proposition 4.1, ∂S,T is nonsingular exactly when T corresponds to a simplicialspanning tree of ∆. Hence Proposition 4.2 gives

detLS =∑

T∈Td(∆)

(|Hd−1(∆T )| · |Hd−2(∆U )|

|Hd−2(∆)|

)2

=|Hd−2(∆U )|2

|Hd−2(∆)|2∑

T∈Td(∆)

|Hd−1(∆T )|2 =|Hd−2(∆U )|2

|Hd−2(∆)|2hd(∆),

which is equivalent to the desired formula.

Remark 4.6. Suppose that Hd−2(∆) = 0. (This occurs, for example, if ∆ is Cohen-Macaulay.) Thenthe two versions of the theorem assert that

hd =πdhd−1

=detLS

|Hd−2(∆U )|2,

from which it is easy to recognize the two different versions of the classical matrix-tree theorem forgraphs. (Note that a graph is Cohen-Macaulay as a simplicial complex if and only if it is connected.)Moreover, the recurrence hd = πd/hd−1 leads to an expression for hd as an alternating product ofeigenvalues:

hd =πdπd−2 · · ·πd−1πd−3 · · ·

.


5. Weighted enumeration of simplicial spanning trees

We can obtain much finer enumerative information by labeling the facets of a complex with inde-terminates, so that the invariants hi and πi become generating functions for its simplicial spanningtrees. The proofs of Theorems 4.3 and 4.5 carry over easily to this more general setting.

As before, let ∆d be a pure metaconnected simplicial complex, and let ∂ = ∂∆,d. Introduce anindeterminate xF for each facet F , and let XF = x2

F . For every T ⊆ ∆d, let xT =∏F∈T xF and

let XT = x2T . To construct the weighted boundary matrix ∂ from ∂, multiply each column of ∂ by

xF , where F is the facet of ∆ corresponding to that column. The weighted coboundary ∂∗ is thetranspose of ∂. We can now define weighted versions of Laplacians, the various submatrices of theboundary and coboundary matrices used in Section 4, and the invariants πk and hk, denoting eachby placing a hat over the symbol for its unweighted analogue. Thus πk is the product of the nonzeroeigenvalues of Lud

∆,k−1, and

hk = hk(∆) =∑

Υ∈Tk(∆)

|Hk−1(Υ)|2XΥ.

To recover any unweighted quantity from its weighted analogue, set xF = 1 for all F ∈ ∆d.

Lemma 5.1. Let T ⊂ ∆d and S ⊂ ∆d−1 such that |T | = |S| = fd−βd. Then det ∂S,T = xT det ∂S,Tis non-zero if and only if T ∈ Td(∆) and S ∈ Td−1(∆). In that case,

±det ∂S,T =|Hd−1(∆T )| · |Hd−2(∆S)|

|Hd−2(∆T )|xT =

|Hd−1(∆T )| · |Hd−2(∆S)||Hd−2(∆)|

xT . (9)

Proof. The first claim follows from Proposition 4.1, and the second from Proposition 4.2.

Theorem 5.2 (Weighted Simplicial Matrix-Tree Theorem I—product of non-zero eigen-values). Let ∆ be a d-dimensional, pure, metaconnected simplicial complex. Then1

πd(∆) =hd(∆)hd−1(∆)|Hd−2(∆)|2

.

Proof. The argument is essentially identical to that of Theorem 4.3. Let L = Lud∆,d−1; then, using

the Binet-Cauchy formula and Lemma 5.1, we have

πd =∑

S⊂∆d−1|S|=fd−βd

det LS =∑S

∑T⊂∆d

|T |=|S|

(det ∂∗T,S)(det ∂S,T ) =∑S

∑T

(det ∂S,T )2

=∑

T∈Td(∆)

∑S∈Td−1(∆)

(det ∂S,T )2

=∑

T∈Td(∆)

∑S∈Td−1(∆)

(|Hd−1(∆T )| · |Hd−2(∆S)|

|Hd−2(∆)|

)2

XT =hd(∆)hd−1(∆)|Hd−2(∆)|2

.

Theorem 5.3 (Weighted Simplicial Matrix Tree Theorem II—determinant of reducedLaplacian). Let ∆ be a d-dimensional, pure, metaconnected simplicial complex. Let U ∈ Td−1(∆),let S = U , and let L be the reduced Laplacian Lud

S . Then

hd(∆) =|Hd−2(∆)|2

|Hd−2(∆U )|2det L.

1There are no missing hats on the right-hand side of this formula! Only hd(∆) has been replaced with its weighted

analogue, not hd−1(∆).


Proof. The argument is essentially identical to that of Theorem 4.5, using Lemma 5.1 instead ofProposition 4.2.

6. Shifted complexes and (the start of) counting their spanning trees

[Note: This used to be near the end of the paper. There should be some transition words sayinghow now we’re going to apply what we’ve done so far, to shifted complexes]

A complex Σ on vertex set V is called a near-cone with apex p if it has the following property: ifσ ∈ delp Σ and v ∈ σ, then σ\v ∈ linkp Σ (equivalently, σ\v ∪ p ∈ Σ).

[Note: Perhaps somewhere define “family”.]If A = a1 < a2 < · · · < ak and B = b1 < b2 < · · · < bk are two k-sets of integers, then

A < B in the componentwise partial order if aj ≤ bj for all j. A k-family F is shifted if B ∈ F andA < B together imply A ∈ F ; equivalently, F is shifted if it is an order ideal with respect to <. Asimplicial complex Σ is shifted if Σi is shifted for all i, with the same ordering of vertices for each i.[Give references for all this?] We will not lose any generality in what we do by assuming thatthe vertex set for every shifted complex we encounter is of the form [p, q] := p, p+ 1, . . . , q.

It is easy to see that, if a complex is shifted on vertex set [p, q], then its deletion and link withrespect to vertex p are again each shifted on vertex set [p+ 1, q].2 It is also easy to see that a shiftedcomplex on with initial vertex p is a near-cone with apex p. Bjorner and Kalai [4, Theorem 4.3]showed that the Betti numbers of a shifted complex Σ (indeed, of a near-cone) with initial vertex pare given by

βi(Σ) = |F ∈ Σi : p 6∈ F, F ∪ p 6∈ Σ|. (10)Let Σ be a pure shifted d-dimensional complex with initial vertex p. We would like to use the

simplicial matrix-tree theorem to give a weighted enumeration of simplicial spanning trees of Σ usingthe doubly indexed weighting (13): that is, we wish to compute

hd(Σ) =∑

T∈T (Σ)

|Hd−1(Σ,Z)|2XT ,

whereXT =

∏facets σ∈T

Xσ.

Accordingly, define the finely weighted simplicial boundary map of Σ as the homomorphism ∂ =∂Σ,i : Ci(Σ)→ Ci−1(Σ) given by

∂[σ] =∑j∈σ

ε(j, σ) ↑d−i(xσ)[σ\j] (11)

and similarly define the finely weighted simplicial coboundary map ∂∗ = ∂∗Σ,i+1 : Ci(Σ) → Ci+1(Σ)as

∂∗[σ] =∑j∈V \σ

ε(j, σ ∪ j) ↑d−i−1(xσ∪j)[σ ∪ j]. (12)

These maps do not make the chain groups of Σ into an algebraic chain complex, because ∂∂ and∂∗∂∗ do not vanish in general. However, they are combinatorially significant, because we can applyVersion II of the weighted simplicial matrix-tree theorem to the finely weighted up-down LaplacianLud = ∂d∂

∗d , as follows. Regard this Laplacian as a matrix whose rows and columns are indexed by

the ridges of Σ. Let T be the simplicial spanning tree of Σ(d−1) consisting of all ridges containingvertex p, and let R be the reduced Laplacian obtained from L by deleting the corresponding rows

2This is also true for the deletion and link with respect to any vertex, not just p, but then the resulting vertex set

is no longer a set of consecutive integers. Since we will not have any need to take the deletion and link with respectto any vertex other than p, we won’t worry about that, and instead enjoy the resulting simplicity of specifying the

new minimal vertex of the deletion and link.


and columns, so that the remaining rows and columns are indexed by the facets of Λ := linkp Σ.Then Version II of the weighted simplicial matrix-tree theorem asserts that

hd(Σ) = detR.

Consider the matrix N obtained from R by dividing each entry in row σ and column τ by themonomial ↑(xσ) ↑(xτ ). Then

detR =

∏σ∈linkp Σ

↑Xσ

detN.

We now claim that N = X1,pI + ∂d∂∗d, where I is an appropriately sized identity matrix and

∂d,∂∗d are the vertex weighted boundary and coboundary maps. Given the claim, it follows that

detN = χ(−M,X1,p) =∏

eigenvalues λ of M

(X1,p + λ),

where χ denotes the characteristic polynomial of −M in the variable X1,p.Note that M = ∂d∂

∗d = Lud

∆,d−1, where ∆ := delp Σ.[Note: The above claim, and all the other missing details of the first part of this section are

discussed in reduction.tex, although in a breezier tone than we’ll want in the paper. I’ll work onrewriting that file to include in here – Art]

We will spend most of the rest of the paper computing the eigenvalues of M , in order to determinehd(Σ).

7. The vertex weighting

We now study a particular weighting by Laurent monomials, defined with respect to a totalordering of the vertices of ∆. This weighting allows us to keep track of the individual verticesand the order in which they appear in faces, while maintaining the structure of an algebraic chaincomplex. It is easy to translate between the vertex weighting and a natural combinatorial weightingof the sort treated in Section 5. Moreover, the vertex weighting behaves well with respect to conesand near-cones, providing a valuable tool in describing the finely weighted Laplacian eigenvalues ofshifted complexes and thus in enumerating their simplicial spanning trees.

Let xi,j be a lot of indeterminates. Let k be the field of rational functions in the xi,j withcoefficients in C (or any other field of characteristic zero). Since these indeterminates will oftenappear squared, we set Xi,j = x2

i,j . Let ↑ be the “raising operator”, defined by ↑xi,j = xi+1,j fori ≤ d and ↑(xd+1,j) = 0, extended linearly and multiplicatively to an operator on all of k. Theraising operator can also be applied to a k-linear operator f by the rule (↑f)(V ) =↑(f(↑−1(V ))),where V denotes any vector over k. We write ↑a for the ath iterate of ↑.

For a multiset of vertices S = v1 ≤ v2 ≤ · · · ≤ vm ∈ ∆i, define monomialsxS = x1,v1x2,v2 · · ·xm,vm ,XS = X1,v1X2,v2 · · ·Xm,vm .

(13)

We point out some useful identities involving the raising operator. Let S = S∪1 (where ∪ denotesthe union as multisets, so that the multiplicity of 1 in S is one more than its multiplicity in S).Then, for all integers p, j,

↑p(x1,1)· ↑p+1(xS∪j) = ↑p(x1,1· ↑xS∪j

)= ↑p(xS∪j) (14a)

and↑a(xS)↑a+1(xS)

= xa+1,1 = ↑a(x1,1). (14b)

The same identities hold if x is replaced with X.Until further notice, let ∆ be a (pure) d-dimensional simplicial complex on V ⊂ N. Let ∆i denote

the set of i-dimensional faces of ∆. Let Ci = Ci(∆) be the ith simplicial chain group, that is, the


k-vector space with basis [σ] : σ ∈ ∆i. For a vertex v and a face σ, let ε(v, σ) = (−1)j+1 if v isthe jth smallest vertex of σ, and ε(v, σ) = 0 if v 6∈ σ.

Definition 7.1. The vertex-weighted simplicial boundary map of ∆ is the homomorphism ∂∆,i :Ci(∆)→ Ci−1(∆) given by

∂∆,i[σ] =∑j∈σ

ε(j, σ)↑d−i(xσ)↑d−i+1(xσ\j)

[σ\j] (15)

and similarly the vertex-weighted simplicial coboundary map ∂∗∆,i+1 : Ci(∆)→ Ci+1(∆) is given by

∂∗∆,i+1[σ] =∑j∈V \σ

ε(j, σ ∪ j)↑d−i−1(xσ∪j)↑d−i(xσ)

[σ ∪ j]. (16)

We will sometimes drop one or both subscripts when stating a general property of these operators.The vertex-weighted boundary and coboundary operators can be represented as matrices with

entries in k; for instance, ∂∆,i has fi−1(∆) rows and fi(∆) columns. Therefore, we can apply theraising operator ↑to ∂ and ∂∗ by applying it to each matrix entry. That is, if [σ] ∈ Ci(∆) and a ∈ N,then

↑a∂∆,i[σ] =∑j∈σ

ε(j, σ)↑d−i+a(xσ)↑d−i+a+1(xσ\j)

[σ\j], (17)

↑a∂∗∆,i[σ] =∑j∈V \σ

ε(j, σ ∪ j)↑d−i+a−1(xσ∪j)↑d−i+a(xσ)

[σ ∪ j]. (18)

The vertex-weighted boundary and coboundary operators induce chain complexes

· · · → Ci+1(∆)∂∆,i+1−−−−→ Ci(∆)

∂∆,i−−−→ Ci−1(∆)→ · · ·

· · · ← Ci+1(∆)∂∗∆,i+1←−−−− Ci(∆)

∂∗∆,i←−−− Ci−1(∆)← · · ·as we now show.

Lemma 7.2. Let a ∈ N. Then ↑a∂ ↑a∂ = 0 and ↑a∂∗ ↑a∂∗ = 0.

Proof. Since the matrices that represent the maps ∂ and ∂∗ are mutual transposes, it suffices toprove the first assertion. For σ ∈ ∆i, we have by (17)

↑a∂∆,i−1(↑a∂∆,i([σ])) =↑a∂∆,i−1

∑j∈σ


[σ\j]

=∑j∈σ


· ↑a∂∆,i−1([σ\j])

=∑j∈σ


∑k∈σ\j

ε(k, σ\j)↑d−i+a+1(xσ\j)↑d−i+a+2(xσ\j\k)

[σ\j\k]

=∑j∈σ

∑k∈σ\j

ε(j, σ)ε(k, σ\j) ↑d−i+a(xσ)↑d−i+a+2(xσ\j\k)

[σ\j\k]

=∑j,k∈σj 6=k

(ε(j, σ)ε(k, σ\j) + ε(k, σ)ε(j, σ\k)

) ↑d−i+a(xσ)↑d−i+a+2(xσ\j\k)

[σ\j\k]

and it is a standard fact from simplicial homology theory that the parenthesized expression iszero.


(Note: Much of the folowing is applicable to any old weighted boundary map thatsatisfies ∂2 = 0.)

We define the (vertex-weighted) up-down, down-up, and total Laplacians by

Lud∆,i = ∂∆,i+1∂

∗∆,i+1, Ldu

∆,i = ∂∗∆,i∂∆,i, Ltot∆,i = Lud

∆,i + Ldu∆,i.

Each of these is a linear endomorphism of Ci(∆), and is represented by a symmetric matrix, henceis diagonalizable. Let sudi (∆), sdui (∆), and stoti (∆) denote the spectra (multisets of eigenvalues) ofLud

∆,i, Ldu∆,i, and Ltot

∆,i respectively. We will use the abbreviated notation Lud, Ldu, Ltot, sud, sdu, stot

when no confusion can arise.

Proposition 7.3. Let −1 ≤ i ≤ d = dim ∆.(1) The chain group Ci(∆) decomposes as a direct sum

Ci(∆) = Cudi (∆)⊕ Cdu

i (∆)⊕ C0i (∆) (19)

where• Cud

i (∆) has a basis consisting of eigenvectors of Ludi whose eigenvalues are all nonzero,

and on which Ldui acts by zero;

• Cdui (∆) has a basis consisting of eigenvectors of Ldu

i whose eigenvalues are all nonzero,and on which Lud

i acts by zero;• Lud

i and Ldui both act by zero on C0

i (∆).(2) ker(Lud

∆,i) = ker(∂∗∆,i) and ker(Ldu∆i

) = ker(∂∆,i).(3) dimC0

i (∆) = βi(∆), the ith Betti number of ∆.(4) Each of the spectra sudi , sdui , stoti has cardinality fi(∆) as a multiset, and

sdui= sudi−1, and (20a)

stoti= sudi ∪ sdui

= sudi ∪ sudi−1. (20b)

Proof. For assertion (1), note that Ludi Ldu

i = ∂∂∗∂∗∂ = 0 and Ldui Lud

i = ∂∗∂∂∂∗ = 0. Thuswe may simply take Cud

i (∆) and Cdui (∆) to be the spans of the eigenvectors of Lud

i and Ldui with

nonzero eigenvalues.For assertion (2), the operators Lud

i and ∂∗∆,i act on the same space, namely Ci(∆), and theyhave the same rank (this is just the linear algebra fact that rank(MMT ) = rankM for any matrixM). Therefore, their kernels have the same dimension. Clearly ker(Lud

i ) ⊇ ker(∂∗∆,i), so we musthave equality. The same argument shows that ker(Ldu

i ) = ker(∂∆,i).Assertion (3) is proved as follows:

βi(∆) = dim Hi(∆,Q) = (dim ker ∂∆,i)− (dim im ∂∆,i+1)

= (fi − rank ∂∆,i)− (rank ∂∆,i+1)

= dimCi − dimCudi − dimCdu

i = dimC0i .

For assertion (4), first note that dimCi(∆) = fi(∆), and that the matrix representing eachspectrum is symmetric, hence diagonalizable. The identity (20a) is a standard fact in linear algebra,and (20b) is a consequence of the decomposition (19).

8. Vertex-weighting, cones, and near-cones

8.1. Boundary and coboundary operators of cones. Let Γ be the simplicial complex with thesingle vertex 1, and let ∆ be any pure d-dimensional complex on V = [2, n]. For σ ∈ ∆, writeσ = 1 ∪ σ. The corresponding cone is

Ω = 1 ∗∆ = Γ ∗∆ = σ, σ : σ ∈ ∆.We will make use of the identification

Ci(Ω) ∼= (C−1(Γ)⊗ Ci(∆))⊕ (C0(Γ)⊗ Ci−1(∆)).


By (17) and (18), the (raised) boundary and coboundary maps on Γ are given explicitly by

↑a∂Γ[1] = xa+1,1[∅], ↑a∂∗Γ[1] = 0,

↑a∂Γ[∅] = 0, ↑a∂∗Γ[∅] = xa+1,1[1].

Next, we give explicit formulas for the maps ∂Ω,i and ∂∗Ω,i+1. How these maps act on a face ofΩ depends on whether it is of the form σ, for σ ∈ ∆i, or σ, for σ ∈ ∆i−1. Note that in any caseε(1, σ) = 1, and that for all v ∈ σ we have ε(v, σ) = −ε(v, σ). Therefore

∂Ω,i([∅]⊗ [σ]) =∑j∈σ

ε(j, σ)↑d−i+1(xσ)↑d−i+2(xσ\j)

[∅]⊗ [σ\j]

=↑

∑j∈σ

ε(j, σ)↑d−i(xσ)↑d−i+1(xσ\j)

[∅]⊗ [σ\j]

= (id⊗ ↑∂∆,i)([∅]⊗ [σ]), (21a)

∂Ω,i([1]⊗ [σ]) = ε(1, σ)↑d−i+1(xσ)↑d−i+2(xσ)

[∅]⊗ [σ] +∑j∈σ


[1]⊗ [σ\j]

= xd−i+2,1[∅]⊗ [σ]−∑j∈σ

ε(j, σ)xd−i+2,1

xd−i+3,1

↑d−i+2(xσ)↑d−i+3(xσ\j)

[1]⊗ [σ\j]

= xd−i+2,1[∅]⊗ [σ]− xd−i+2,1

xd−i+3,1↑

∑j∈σ


[1]⊗ [σ\j]

=(↑d−i+1∂Γ ⊗ id−xd−i+2,1

xd−i+3,1id⊗ ↑∂∆,i−1

)([1]⊗ [σ]), (21b)

∂∗Ω,i+1([∅]⊗ [σ]) = ε(1, σ)↑d−i(xσ)↑d−i+1(xσ)

[1]⊗ [σ] +∑j∈V \σ

ε(j, σ ∪ j) ↑d−i(xσ∪j)↑d−i+1(xσ)

[∅]⊗ [σ ∪ j]

= xd−i+1,1[1]⊗ [σ]+ ↑

∑j∈V \σ

ε(j, σ ∪ j)↑d−i−1(xσ∪j)↑d−i(xσ)

[∅]⊗ [σ ∪ j]

=(↑d−i∂∗Γ ⊗ id + id⊗ ↑∂∗∆,i+1

)([∅]⊗ [σ]), (21c)

∂∗Ω,i+1([1]⊗ [σ]) =∑j∈V \σ

ε(j, σ ∪ j) ↑d−i(xσ∪j)↑d−i+1(xσ∪j)

[1]⊗ [σ ∪ j]

= −∑j∈V \σ

ε(j, σ ∪ j) ↑d−i(x1,1)↑d−i+1(x1,1)

↑d−i+1(xσ∪j)↑d−i+2(xσ)

[1]⊗ [σ ∪ j]

= −xd−i+1,1

xd−i+2,1· ↑

∑j∈V \σ

ε(j, σ ∪ j) ↑d−i(xσ∪j)↑d−i+1(xσ)

[1]⊗ [σ ∪ j]

=(−xd−i+1,1

xd−i+2,1id⊗ ↑∂∗∆,i

)([1]⊗ [σ]). (21d)


8.2. Eigenvectors of cones. Our next goal is to describe the Laplacian eigenvalues and eigenvec-tors of 1 ∗ Λ in terms of those of Λ.

Theorem 8.1. As before, let Λ be a d-dimensional simplicial complex on V = [2, n], and let ∆ =1 ∗ Λ. Then

sudi (∆) =Xd−i+1,1+ ↑λ : λ ∈ sudi (Λ), λ 6= 0

∪Xd−i+1,1 +

Xd−i+1,1

Xd−i+2,1↑µ : µ ∈ sudi−1(Λ), µ 6= 0

∪ Xd−i+1,1βi(Λ)

.

(22)

Here the symbol ∪ denotes multiset union, and the superscript in the last line of (22) indicatesmultiplicity.

Proof. Throughout the proof, we abbreviate L = Ludi (∆); all other Laplacians that arise will be

specified precisely.First, let V ∈ Cud

i (Λ) be an eigenvector of LudΛ,i with eigenvalue λ 6= 0. Then ↑Lud

Λ,i(↑V ) =↑λ ↑V ,and, by Lemma 7.2,

↑∂Λ,i(↑V ) =1↑λ↑∂Λ,i

(↑∂Λ,i+1(↑∂∗Λ,i+1(↑V ))

)= 0. (23)

Using (21a). . . (21d) and (23), we calculate

L ([∅]⊗ ↑V ) = ∂∆,i(∂∗∆,i([∅]⊗ ↑V ))

= ∂∆,i

(↑d−i∂∗Γ[∅]⊗ ↑V + [∅]⊗ ↑∂∗Λ,i+1(↑V )

)= xd−i+1,1∂∆,i+1 ([1]⊗ ↑V ) + ∂∆,i+1

([∅]⊗ ↑∂∗Λ,i+1(↑V )

)= xd−i+1,1

(↑d−i∂Γ[1]⊗ ↑V − xd−i+1,1

xd−i+2,1[1]⊗ ↑∂Λ,i(↑V )

)+ [∅]⊗ ↑∂Λ,i+1(↑∂∗Λ,i+1(↑V ))

= (Xd−i+1,1+ ↑λ) ([∅]⊗ ↑V ) . (24)

Therefore, [∅]⊗ ↑V ∈ Ci(∆) is an eigenvector of L, with eigenvalue Xd−i+1,1+ ↑λ. Notice that thiseigenvalue cannot be zero: since 1 6∈ Λ, no Laplacian eigenvalue of Λ can possibly equal −Xd−i,1.

Second, let W be an eigenvector in Cdui (Λ) with nonzero eigenvalue µ. That is, Ldu

Λi(W ) =

∂∗Λ,i(∂Λ,i(W )) = µW , and ∂∗Λ,i+1(W ) = 0 by a computation similar to (23).Define

A = [∅]⊗ ↑W, B = [1]⊗ ↑∂Λ,i(↑W );

these are both nonzero elements of Ci(∆). Then

L(A) = ∂∆,i+1

(∂∗∆,i+1([∅]⊗ ↑W )

)= ∂∆,i+1

(↑d−i∂∗Γ[∅]⊗ ↑W + [∅]⊗ ↑∂∗Λ,i+1(↑W )

)= xd−i+1,1∂∆,i+1 ([1]⊗ ↑W ) (25a)

= xd−i+1,1

(↑d−i∂Γ[1]⊗ ↑W − xd−i+1,1

xd−i+2,1[1]⊗ ↑∂Λ,i(↑W )

)= Xd−i+1,1A−

(Xd−i+1,1

xd−i+2,1

)B (25b)


and

L(B) = ∂∆,i+1

(∂∗∆,i+1([1]⊗ ↑∂Λ,i(↑W ))

)= ∂∆,i+1

(−xd−i+1,1

xd−i+2,1[1]⊗ ↑∂∗Λ,i(↑∂Λ,i(↑W ))

)= −xd−i+1,1

xd−i+2,1↑µ · ∂∆,i+1 ([1]⊗ ↑W )

=(−Xd−i+1,1

xd−i+2,1↑µ)A+

(Xd−i+1,1

Xd−i+2,1↑µ)B,

where we have reused the equality between (25a) and (25b) . Letting [uh oh!! Here it will be hardto use p for the first vertex!]

p = Xd−i+1,1, q = −Xd−i+1,1

xd−i+2,1, r = − ↑µ

xd−i+2,1,

the calculations above say that

L(A) = pA+ qB, L(B) = r(pA+ qB),

which implies that

L(pA+ qB) = pL(A) + qL(B) = p(pA+ qB) + qr(pA+ qB) = (p+ qr)(pA+ qB).

That is, pA+ qB is an eigenvector of L with eigenvalue

p+ qr = Xd−i+1,1 +Xd−i+1,1

Xd−i+2,1↑µ.

As before, this quantity cannot be zero because Λ does not contain the vertex 1.Third, let Z ∈ C0

i (Λ); we will show that [∅]⊗ ↑Z is an eigenvector of L with eigenvalue Xd−i+1,1.Indeed, by (2) of Proposition 7.3, we have ∂Λ,i(Z) = ∂∗Λ,i+1(Z) = 0, so that

↑∂Λ,i(↑Z) =↑∂∗Λ,i+1(↑Z) = 0.

Therefore

L([∅]⊗ ↑Z) = ∂∆,i+1(∂∗∆,i+1([∅]⊗ ↑Z))

= ∂∆,i+1

((↑d−i∂∗Γ[∅]⊗ ↑Z) + [∅]⊗ ↑∂∗Λ,i+1(↑Z)

)= xd−i+1,1∂∆,i+1([1]⊗ ↑Z)

= xd−i+1,1 ·(↑d−i∂Γ[1]⊗ ↑Z − xd−i+1,1

xd−i+2[1]⊗ ↑∂Λ,i(↑Z)

)= Xd−i+1,1([∅]⊗ ↑Z), (26)

as desired. By assertion (3) of Proposition 7.3, the multiplicity of this eigenvalue is dimC0i (Λ) =

βi(Λ),At this point, we have proven that (22) is true if “ =” is replaced with “⊇”. On the other hand,

we have accounted for

dimCudi (Λ) + dimCdu

i (Λ) + dimC0i (Λ) = dimCi(Λ) = fi(Λ)

nonzero eigenvalues in sudi (∆). Since the preceding calculations hold for all i, we also know fi−1(Λ)nonzero eigenvalues of Lud

∆,i−1 ( = Ldu∆,i). By (20b), we have accounted for all fi(Λ) + fi−1(Λ) =

fi(∆) = dimCi(∆) eigenvalues of Ltot∆,i. So we have indeed found all the nonzero eigenvalues of

L.

One can obtain explicit formulas for the spectra sdui (∆) and stoti (∆) by applying (20a) and (20b)to the formula (22); we omit the details.


8.3. The vertex weighting and near-cones. In this section, we mimic the setup and proof ofLemma 5.3 of [9] to obtain a recurrence, Proposition 8.5, for the Laplacian eigenvalues of near-conesin terms of the eigenvalues of the deletion and link of the apex. Note that, by itself, Proposition 8.5is not a proper recurrence, per se, in that it only computes the eigenvalues of a pure complex in termsof not necessarily pure complexes, and so it cannot be applied recursively. The proper recurrencefor shifted complexes will have to wait until later! – put in proper reference once structureof paper is more firmed up.

Since we will be comparing complexes with similar face sets but of different dimensions, we beginby describing how their Laplacian spectra are related. We will need the “lowering operator” ↓ definedas the inverse of ↑: that is, ↓ xi,j = xi−1,j , extended to a ring endomorphism on k. If S is a set ofrational functions, we write ↓ S for ↓ f : f ∈ S. [Question: Why not just write Lemma 8.2 assudj (∆) =↑d−isudj (∆(i))?]

Lemma 8.2. Let ∆ be a simplicial complex of dimension d, and let j < i ≤ d. Then

sudj (∆(i)) =↓d−i sudj (∆).

Proof. The complexes ∆(i) and ∆ have the same face sets for every dimension ≤ i, but dim ∆(i) =dim ∆ − (d − i). Therefore, the vertex-weighted boundary maps and Laplacians of ∆(i) can beobtained from those of ∆ by applying ↓d−i, from which the lemma follows.

[Note: Maybe Lemma 8.3 and Corollary 8.4 should change to ∆, not Σ. Also need a reminderof definition of pure skeleton.]

Lemma 8.3. If Σd is a d-dimensional simplicial complex, then

sudd−1(Σ) = sudd−1(Σ[d]).

Proof. This result is proved in [8, Lemma 3.2], but we sketch the proof here for completeness. Firstnote that Lud

d−1 depends only on (d − 1)- and d-dimensional faces. In these two dimensions, Σ andΣ[d] differ only in some (possible) additional (d − 1)-dimensional faces of Σ that are not containedin any d-dimensional faces of Σ. As a result, for each of these faces σ, we have Lud

Σ,d−1σ = 0, andfor all other (d− 1)-dimensional faces τ of both complexes, Lud

Σ,d−1τ = LudΣ[d],d−1τ . The lemma now

follows readily.

Corollary 8.4. If Σd is a d-dimensional simplicial complex, then

sudi−1(Σ) =↑d−isudi−1(Σ[i]).

Proof. By Lemmas 8.2 and 8.3,

sudi−1(Σ) =↑d−isudi−1(Σ(i))=↑d−isudi−1((Σ(i))[i]) =↑d−isudi−1(Σ[i]),

since (Σ(i))[i] = Σ[i].

Proposition 8.5. Let Σd be a pure near-cone with apex p, and let ∆ = delp Σ and Λ = linkp Σ bethe deletion and link, respectively, of Σ with respect to vertex p. Then

sudd−1(Σ) =X1,p + λ : λ ∈ sudd−1(∆), λ 6= 0

∪X1,p +

X1,p

X2,p↑µ : µ ∈ sudd−2(Λ), µ 6= 0

∪ X1,pβd−1(∆)

.

Proof. If Σ is a cone, then the proposition follows from direct application of Theorem 8.1. (Notethat sudd−1(∆) consists of only 0’s in this case, since ∆ is only (d − 1)-dimensional.) Thus, we mayas well assume for the remainder of the proof that Σ is not a cone. In this case, dim ∆ = d anddim Λ = d− 1. We now claim that

∆(d−1) = Λ. (27)


To prove this claim, first note that Λ ⊆ ∆ is obvious, and so Λ ⊆ ∆(d−1), since dim Λ = d−1. Next,if F ∈ ∆(d−1), then p 6∈ F , F ∈ Σ, and dimF ≤ d− 1. By purity, F is in some d-dimensional face,so F ∪ v ∈ Σ for some v ∈ V . Since F ∪ v ∈ Σ and Σ is a near-cone, F ∪ p ∈ Σ, so F ∈ Λ, whichestablishes the claim.

We next claim that

Σ(d) = Σ = (p ∗∆)(d). (28)

To show that Σ ⊆ (p ∗∆)(d), it suffices to show Σ ⊆ (p ∗∆), since Σ is only d-dimensional. To thisend, let F ∈ Σ. If F ∈ ∆, we are done, so assume F 6∈ ∆, which means p ∈ F . We may thus letF = F ′ ∪ p for some F ′, but this immediately shows F ∈ p ∗∆.

To show that (p ∗∆)(d) ⊆ Σ, let F ∈ (p ∗∆)(d). If F ∈ ∆, we are done, since ∆ ⊆ Σ, so assumeF 6∈ ∆. Then F = F ′ ∪ p for some F ′ ∈ ∆, and dimF ′ ≤ d − 1 since dimF ≤ d. By purity then,F ′ ∪ v ∈ Σ for some v ∈ V . Since F ′ ∪ v ∈ Σ and Σ is a near-cone, F = F ′ ∪ p ∈ Σ. This establishesthe second claim.

Therefore, by Lemma 8.2 applied to equation (28) (keeping in mind dim p ∗∆ = d+ 1), and thenby Theorem 8.1,

sudd−1(Σ) =↓ sudd−1(p ∗∆)=↓X2,p+ ↑λ : λ ∈ sudd−1(∆), λ 6= 0

∪ ↓

X2,p +

X2,p

X3,p↑µ : µ ∈ sudd−2(∆), µ 6= 0

∪ ↓ X2,pβd−1(∆)

=X1,p + λ : λ ∈ sudd−1(∆), λ 6= 0

∪X1,p +

X1,p

X2,pµ : µ ∈ sudd−2(∆), µ 6= 0

∪ X1,pβd−1(∆)

.

The proposition now follows from Lemma 8.2 again, this time applied to equation (27).

[Note: This would be the place for “Corollary 9.3”]

9. z-polynomials and critical pairs

9.1. z-polynomials. Let S and T be multisets of integers. Define a Laurent polynomial z(S, T ) bythe formula

z(S, T ) =1↑XS

∑j∈T

XS∪j , (29)

where as usual the symbol ∪ denotes multiset union.

Proposition 9.1. Let d > i be integers, and let S, T be sets of integers greater than p. Then thefollowing identities hold:

Xd−i+1,p+ ↑d−iz(S, T ) =↑d−iz(S, T ) (30)

and

Xd−i+1,p +Xd−i+1,p

Xd−i+2,p↑d−i+1z(S, T ) =↑d−iz(S, T ) (31)

where S = S ∪ p, T = T ∪ p.


Proof. We will use the identity (14a) [or some generalization thereof, to deal with moregeneral indices] repeatedly in the calculations. First,

Xd−i+1,p+ ↑d−iz(S, T ) = Xd−i+1,p +1

↑d−i+1XS

∑j∈T↑d−iXS∪j

=1

↑d−i+1XS

Xd−i+1,p· ↑d−i+1XS +∑j∈T↑d−iXS∪j

=

1↑d−i+1XS

↑d−iXS +∑j∈T↑d−iXS∪j

=

1↑d−i+1XS

∑j∈T

↑d−iXS∪j

=↑d−iz(S, T ),

and second,

Xd−i+1,p +Xd−i+1,p

Xd−i+2,p↑d−i+1z(S, T ) = Xd−i+1,p +

Xd−i+1,p

Xd−i+2,p· ↑d−i+2XS

∑j∈T↑d−i+1XS∪j

= Xd−i+1,p +Xd−i+1,p

↑d−i+1XS

∑j∈T↑d−i+1XS∪j

=1

↑d−i+1XS

Xd−i+1,p· ↑d−i+1XS +∑j∈T

Xd−i+1,p· ↑d−i+1XS∪j

=

1↑d−i+1XS

↑d−iXS∪p +∑j∈T↑d−iXS∪j

=

1↑d−i+1XS

∑j∈T

↑d−iXS∪j

=↑d−iz(S, T ).

Theorem 9.2. Let Σd be a shifted simplicial complex. The nonzero eigenvalues of sudi−1(Σ) aredetermined recursively as follows.

• Every nonzero eigenvalue of sudi−1(Σ) has the form of a z-polynomial.• If Σ has no vertices, then sudi−1(Σ) consists only of 0’s.• Otherwise,

sudi−1(Σ) =↑d−iz(S, T ) : 0 6= z(S, T ) ∈ sudi−1(∆)

∪↑d−iz(S, T ) : 0 6= z(S, T ) ∈ sudi−2(Λ)

∪↑d−iz(∅, ∅)

βi−1(∆)

,

(32)

where p is the initial vertex of Σ[i], S = S ∪ p, T = T ∪ p, ∆ = delp Σ[i], and Λ = linkp Σ[i].

Proof. The proof is by induction on the number of vertices of Σ. When Σ has no vertices, Σ is eitherthe empty simplicial complex with no faces, or the simplicial complex whose only face is the emptyface. In either case, it is not hard to see that sudi−1(Σ) consists only of 0’s.


So now we may assume that Σ has at least one vertex, and then

sudi−1(Σ[i])=X1,p + λ : λ ∈ sudi−1(∆), λ 6= 0

∪X1,p +

X1,p

X2,p↑µ : µ ∈ sudi−2(Λ), µ 6= 0

∪ X1,p + 0βi−1(∆)

=X1,p + z(S, T ) : z(S, T ) ∈ sudi−1(∆), z(S, T ) 6= 0

∪X1,p +

X1,p

X2,pz(S, T ) : z(S, T ) ∈ sudi−2(Λ), z(S, T ) 6= 0

∪ X1,p + z(∅, ∅)βi−1(∆)

=z(S, T ) : 0 6= z(S, T ) ∈ sudi−1(∆)

∪z(S, T ) : 0 6= z(S, T ) ∈ sudi−2(Λ)

∪z(∅, ∅)

βi−1(∆)

.

The =-equivalence above is by Proposition 8.5. The equality following it uses the identity z(∅, ∅) = 0and induction, since ∆ and Λ each have one fewer vertex than Σ. Note that λ and µ are eachreplaced by z(S, T ), with no ↑operator, because dim ∆ ≤ i and dim Λ = i− 1. The final equality isby Proposition 9.1. The result now follows from Corollary 8.4.

9.2. Critical pairs.

Definition 9.3. A critical pair for a k-family F is an ordered pair (A,B), where A = a1 < a2 <· · · < ak and B = b1 < b2 < · · · < bk are sets of integers such that A ∈ F , B 6∈ F , and B coversA in componentwise order. That is, bi = ai + 1 for exactly one i, and bj = aj for all j 6= i. Notethat bi might not be in the vertex set of F .

The signature σ(A,B) of (A,B) is the set of vertices a1, . . . , ai−1, ai. The long signatureτp(A,B) is the ordered pair of sets (S, T ) where S = a1, . . . , ai−1 and T = j : p ≤ j ≤ ai.

The set of critical pairs is especially significant for a shifted family (and by extension, for ashifted complex). Since a shifted family is an order ideal with respect to <, members of the familylie below the critical pairs, non-members lie above, and the critical pairs themselves identify theexact boundary between the two in the Hasse diagram of <. [Note: Possibly a reference to Carly’s“Obstructions to shiftedness” paper may be nice here, especially if we can tie in to “critical pairs”where 0 is replaced by 1.]

[Note: Maybe use v instead of j for typical vertex below.]

Definition 9.4. The degree of vertex j in family F is

degF (j) = |F ∈ F : j ∈ F|.

Proposition 9.5. If F is a shifted family, then degF (j)−degF (j+1) counts the number of signaturesof F whose greatest element is j.

This proposition is used in the recursion below, and will also later! show how the results in thissection specialize to the more familiar formula for unweighted eigenvalues in the obvious way.


Proof. Let S = F ∈ F : j ∈ F and T = F ∈ F : j + 1 ∈ F. Partition S = S1 ∪S2 ∪S3 andT = T1 ∪T2 ∪T3 as follows:

S1 = F ∈ F : j, j + 1 ∈ FS2 = F ∈ F : j ∈ F, j + 1 6∈ F, F − j ∪ (j + 1) ∈ FS3 = F ∈ F : j ∈ F, j + 1 6∈ F, F − j ∪ (j + 1) 6∈ F

T1 = F ∈ F : j, j + 1 ∈ FT2 = F ∈ F : j 6∈ F, j + 1 ∈ F, F − (j + 1) ∪ j ∈ FT3 = F ∈ F : j 6∈ F, j + 1 ∈ F, F − (j + 1) ∪ j 6∈ F.

Then S1 = T1, and there is an obvious bijection between S2 and T2. Since F is shifted, T3 = ∅, sodegF (j)− degF (j + 1) = |S| − |T | = |S3|.

Now note that if F ∈ S3, then (F,G) is a critical pair, where G = F − j ∪ (j + 1). In each suchcritical pair, the greatest element of the signature is j. Conversely, if (A,B) is a critical pair whosesignature’s greatest element is j, then A ∈ S3.

Later, we’ll show the specialization hinted at above (before the proof). For now, a quick corollarywe’ll need shortly.

Corollary 9.6. If Σ is a pure d-dimensional shifted simplicial complex with initial vertex p, thendegΣd

(p)− degΣd(p+ 1) = βd−1(delp Σ).

Proof. As in the proof of Proposition 9.5 above,

degΣd(p)− degΣd

(p+ 1) = |S3|,and, in this case,

S3 = F ∈ Σd : p ∈ F, p+ 1 6∈ F, F − p ∪ (p+ 1) 6∈ Σd.There is an obvious bijection between S3 and

S′3 = G ∈ (delp Σ)d−1 : p+ 1 6∈ G, G ∪ (p+ 1) 6∈ delp Σgiven by G = F − p. Then, by equation (10),

βd−1(delp Σ) = |S′3|.Combining the four displayed equations yields the corollary.

[Note: We need to define σ acting on a whole family, in the obvious way.]

Proposition 9.7. Let Σ be a shifted complex with initial vertex p. Let ∆ = delp Σ and Λ = linkp Σ.Then

σ(Σi) = σ(∆i) ∪ p ∪F : F ∈ Λi−1 ∪ pdegΣi(p)−degΣi

(p+1),

where ∪ denotes multiset union.

Proof. First note that Proposition 9.5 immediately implies that the signature p occurs with thedesired multiplicity.

If A ∈ ∆i, B 6∈ ∆i, and B covers A in the componentwise partial order, then A ∈ Σi and p 6∈ A.We will show B 6∈ Σi. But since p 6∈ A and A < B, we conclude p 6∈ B, and so B 6∈ ∆i impliesB 6∈ Σi. This establishes the multiset injection σ(∆i)→ σ(Σi).

If A ∈ Λi−1, B 6∈ Λi−1, and B covers A in the componentwise partial order, then p 6∈ A andp ∪A ∈ Σi. We will show that p ∪B 6∈ Σi; thus the critical pair (A,B) in Λi−1 gives rise to thecritical pair (p ∪A, p ∪B) in Σi, and it is clear σ(p ∪A, p ∪B) = p ∪σ(A,B). Since A < B, thenp ∪A < p ∪B. Furthermore, B 6∈ Λi−1 implies p ∪B 6∈ Σi. This establishes the multiset injectionp ∪F : F ∈ σ(Λi−1) → σ(Σi).

Now we must show, conversely, that every signature F 6= p of Σi arises from σ(∆i) orp ∪F : F ∈ σ(Λi−1) as described above. Let F 6= p be a signature of Σi, with critical pair


(A,B), so A ∈ Σi, B 6∈ Σi, and B covers A in the componentwise partial order. If p 6∈ F , then p 6∈ Aand so A ∈ ∆i. Also B 6∈ ∆i, since B 6∈ Σi. Thus (A,B) is a critical pair for ∆i.

Otherwise, p ∈ F , so F = F ′ ∪ p where F ′ 6= ∅. In this case, we claim that p ∈ A and p ∈ B forthe critical pair (A,B) whose signature is F : p ∈ F directly implies p ∈ A; and then p ∈ B, sincep 6∈ B and p ∈ A would lead to the contradiction F = σ(A,B) = p. Thanks to this claim, wemay let A = p ∪A′ and B = p ∪B′. From A ∈ Σi it follows A′ ∈ Λi−1, and similarly B 6∈ Σi impliesB′ 6∈ Λi−1. Thus (A,B) is a critical pair for Λi−1.

If family F has initial vertex p, then denote by ψ(F) the multiset of long signatures, with respectto p, of critical pairs of F . In other words, ψ(F) = τp(A,B) : (A,B) is a critical pair of F.

Corollary 9.8. Let Σ be a shifted simplicial complex. The multiset of long signatures τp of itsi-dimensional critical pairs are determined recursively as follows.

• If Σ has no vertices, then ψ(Σi) = ∅.• Otherwise,

ψ(Σi) = (S, T ) : (S, T ) ∈ ψ(∆i) ∪ (S, T ) : (S, T ) ∈ ψ(Λi−1) ∪ (∅, ∅)βi−1(∆),

where p is the initial vertex of Σ[i], S = S ∪ p, T = T ∪ p, ∆ = delp Σ[i], and Λ = linkp Σ[i].

Proof. For the third set of the right-hand side, note that if σ(A,B) = p, then τp(A,B) = (∅, p) =(∅, ∅), and apply Corollary 9.6 to the pure i-dimensional shifted complex Σ[i].

For the first two sets, the only hard part is to note that if the first vertex of Σi is p, then p+ 1 isthe first vertex of ∆i and Λi−1 (unless ∆i = ∅, but in that case we don’t have to worry about thefirst set). This accounts for the T ’s.

Even though Proposition 9.7 above defines ∆ and Λ somewhat differently than here, it is not aproblem because it is almost obvious that (delp Σ)i = (delp Σ[i])i and (linkp Σ)i−1 = (linkp Σ[i])i−1.Then, since ∆ and Λ only appear as ∆i and Λi−1, it doesn’t matter whether we set them to be thedeletion and link of Σ or of Σ[i].

[Note: This proof feels a little sketchy, but I’m not sure what else to say, or how to say it. Atleast not yet.]

[Somewhere: note that Σ[i] is shifted when Σ is, or even just when Σi is.]

9.3. The theorem, and its consequences.

Theorem 9.9. Let Σd be a shifted simplicial complex. Then, for each 0 ≤ i ≤ d,

sudi−1(Σ) =↑d−iz(S, T ) : (S, T ) ∈ ψ(Σi).

Proof. “Here see ye two recurrences, and lo! they are the same.” (Theorem 9.2 and Corollary 9.8.)

Alternate proof. The proof is by induction on the number of vertices of Σ. When Σ has no vertices,Σ is either the empty simplicial complex with no faces, or the simplicial complex whose only face isthe empty face. In either case, it is not hard to see that both sides of the =-equivalence are empty.


So now we may assume that Σ has at least one vertex. We will first work with the pure i-skeletonΣ[i]. Assume the initial vertex of Σ[i] is p. Let ∆ = delp Σ[i] and Λ = linkp Σ[i]. Then

sudi−1(Σ[i])=X1,p + λ : λ ∈ sudi−1(∆), λ 6= 0

∪X1,p +

X1,p

X2,p↑µ : µ ∈ sudi−2(Λ), µ 6= 0

∪ X1,p + 0βi−1(∆)

= X1,p + z(S, T ) : (S, T ) ∈ ψ(∆i)

∪X1,p +

X1,p

X2,p↑z(S, T ) : (S, T ) ∈ ψ(Λi−1)

∪ X1,p + z(∅, ∅)βi−1(∆)

=z(S, T ) : (S, T ) ∈ ψ(∆i)

∪z(S, T ) : (S, T ) ∈ ψ(Λi−1)

∪z(∅, ∅)

βi−1(∆)

=z(S, T ) : (S, T ) ∈ ψ((Σ[i])i)

.

The =-equivalence above is by Proposition 8.5. The first equality (after the =-equivalence) usesthe identity z(∅, ∅) = 0 and induction, since ∆ and Λ each have one fewer vertex than Σ, and sincedim ∆ = i and dim Λ = i− 1. (More precisely, dim ∆ ≤ i, but if dim ∆ < i, then sudi−1(∆) is empty.)The next equality is by Proposition 9.1. The final equality is by Proposition 9.8 with Corollary 9.6.We now conclude, by Corollary 8.4, that

sudi−1(Σ) =↑d−isudi−1(Σ[i])=↑d−i

z(S, T ) : (S, T ) ∈ ψ((Σ[i])i)

=↑d−i z(S, T ) : (S, T ) ∈ ψ(Σi) .

Here we show how Theorem 9.9 specializes to the previously known formula from [9], and ageneralization of that formula to coarse weighting.

Definition 9.10. If λ is a partition, let Xλ be the multiset where each part i of λ is replaced byX〈i〉 := X1 + · · ·+Xi.

Remark 9.11. It is standard partition theory (isn’t it?) that λ′, the conjugate of partition λ maybe described as the partition whose part t occurs with multiplicity λt − λt+1.

Corollary 9.12. In the coarse weighting,

sudd−1(Σ) = X(degΣd)′ ,

where Σ is a shifted d-dimensional complex with initial vertex 1, and degΣddenotes the partition of

its degree sequence

Proof. To reduce to coarse weighting, set Xi,j = Xj , so

z(S, T ) =∑j∈T

Xj = X1 + · · ·+Xt = X〈t〉

where T = [1, t]. In this case, if (A,B) is a critical pair of Σd, and if τ1(A,B) = (S, T ), then z(S, T ) =X〈t〉, where t = maxσ(A,B). By Theorem 9.9, then, sudd−1(Σ) is =-equivalent to the partitionwhere X〈t〉 occurs with multiplicity |(A,B) : (A,B) is a critical pair of Σd, t = maxσ(A,B)|. ByProposition 9.5 that multiplicity may be counted by degΣd

(t) − degΣd(t + 1). The corollary now

follows from Remark 9.11


The formula in [9] follows immediately from this corollary.

9.4. An example: the bipyramid. To illustrate Theorem 8.1 and Proposition 8.5, [just Propo-sition 8.5, I think] we calculate the top-dimensional up-down Laplacian spectrum for the bipyramidcomplex B = 〈123, 124, 125, 134, 135, 234, 235〉. The bipyramid is shifted, which is equivalent to thecondition that it is an iterated near-cone. [Citation?]

In our recursive calculation, we will encounter the following subcomplexes of B:

B1 = B, B5 = 〈3, 4, 5〉,B2 = 〈234, 235〉, B6 = 〈4, 5〉,B3 = 〈23, 24, 25, 34, 35〉, B7 = 〈5〉,B4 = 〈34, 35〉, B8 = ∅.

3

5

2

4

5

2

3

1

4

2 3

4

5

B1 B2 B3 B4 B5 B6

3

4

5

4

5

3

4

5

[Notes for diagram: Can we add B7? Can we make 3 the middle vertex in B4 and B5? (In B4, thisis necessary, so then it would be nice to change B5 for consistency.) In B1, the center triangle needsto be vertices 1,2,3. For consistency, I would recommend putting 4 at the top, 5 on the bottom, 2on the left, 1 in the (lower) right and 3 in the (upper) right.]

Observe that

• B1 is a near-cone with apex 1, del1B1 = B2, and link1B1 = B3;• B2 = 2 ∗B4;• B3 is a near-cone with apex 2, del2B3 = B4, and link2B3 = B5;• B4 = 3 ∗B6;• B5 is a near-cone with apex 3, del3B5 = B6, and link3B5 = B8;• B6 is a near-cone with apex 4, del4B6 = B7, and link4B6 = B8; and• B7 = 4 ∗B8.

[Note: We might get rid of the calculation of eigenvalues without z’s.]First, B8 has no vertices, and has only a 0 eigenvalue.Second, to calculate sud−1(B7), we apply Theorem 8.1 with Σ = B7, Λ = B8, and d = i = −1

[or Proposition 8.5 with Σ = B7, ∆ = Λ = B8 and d = 0]. Note that sud−2(Λ) = sud−1(Λ) = ∅ andβ−1(Λ) = 1, [for Proposition 8.5, note that sud−2(Λ) = sud−1(∆) = ∅ and β−1(∆) = 1] so we obtain

sud−1(B7) = X151. (33)

Third, to calculate sud−1(B6), we apply Proposition 8.5 with Σ = B6, ∆ = B7, Λ = B8, and d = 0.Note that sud−2(Λ) = ∅ and β−1(∆) = 0, so we obtain

sud−1(B6) =X14 + λ : λ ∈ sud−1(B7), λ 6= 0

= X14 +X15 . (34)


Fourth, to calculate sud−1(B5), we apply Proposition 8.5 with Σ = B5, ∆ = B6, Λ = B8, and d = 0.Note that sud−2(Λ) = ∅ and that β−1(∆) = 0, so we obtain

sud−1(B5) =X13 + λ : λ ∈ sud−1(B6), λ 6= 0

= X13 +X14 +X15 . (35)

Fifth, to calculate sud0 (B4), we apply Theorem 8.1 with Λ = B6, Σ = B4, and d = i = 0. [orProposition 8.5 with Σ = B4, ∆ = Λ = B6, and d = 0.] Note that sud0 (Λ) = ∅ and β0(Λ) = 1, [forProposition 8.5, note that sud0 (∆) = ∅ and β0(∆) = 1] so we obtain

sud0 (B4) =X13 +

X13

X23↑µ : µ ∈ sud−1(B6), µ 6= 0

∪ X131

=X13(X23 +X24 +X25)

X23, X13

. (36)

Sixth, to calculate sud0 (B3), we apply Proposition 8.5 with Σ = B3, ∆ = B4, Λ = B5, and d = 1.Note that β0(∆) = 0. Thus

sud0 (B3) =X12 + λ : λ ∈ sud0 (B4), λ 6= 0

∪X12 +

X12

X22↑µ : µ ∈ sud−1(B5), µ 6= 0

=X12 +

X13(X23 +X24 +X25)X23

, X12 +X13

∪X12 +

X12

X22↑(X13 +X14 +X15)

=X12X23 +X13X23 +X13X24 +X13X25

X23, X12 +X13,

X12(X22 +X23 +X24 +X25)X22

.

(37)

Seventh, to calculate sud1 (B2), we apply Theorem 8.1 with Λ = B4, Σ = B2, and d = i = 1. [orProposition 8.5 with Σ = B2, ∆ = Λ = B4, and d = 2.] Note that s1(Λ) = ∅ and β1(Λ) = 0, so [forProposition 8.5, note s1(∆) = ∅ and β1(∆) = 0]

sud1 (B2) =X12 +

X12

X22↑µ : µ ∈ sud−1(B5), µ 6= 0

=X12 +

X12

X22↑(X13(X23 +X24 +X25)

X23

), X12 +

X12

X22↑(X13)

=X12(X22X33 +X23X33 +X23X34 +X23X35)

X22X33,X12(X22 +X23)

X22

. (38)

Finally, we calculate sud1 (B) by applying Proposition 8.5 with Σ = B, ∆ = B2, Λ = B3, andd = 2. Note that β1(∆) = 0, so

sud1 (B) =X11 + λ : λ ∈ sud1 (B2), λ 6= 0

∪X11 +

X11

X21↑µ : µ ∈ sud0 (B3), µ 6= 0

= (X11X22X33 +X12X22X33 +X12X23X33 +X12X23X34 +X12X23X35

X22X33,

X11X22 +X12X22 +X12X23

X22,

X11(X21X33 +X22X33 +X23X33 +X23X34 +X23X35)X21X33

,

X11(X21 +X22 +X23)X21

,

X11(X21X32 +X22X32 +X22X33 +X22X34 +X22X35)X21X32

. (39)


Example 9.13. The notation of z-polynomials makes it much easier to calculate the nonzero Lapla-cian eigenvalues of the bipyramid B = B1 and its subcomplexes B2, . . . , B7:

sud−1(B7) = z(∅, 5),

sud−1(B6) = z(∅, 45),

sud−1(B5) = z(∅, 345),

sud0 (B4) = z(3, 345), z(∅, 3),

sud0 (B3) = z(3, 2345), z(∅, 23), z(2, 2345),

sud1 (B2) = z(23, 2345), z(2, 23),

sud1 (B1) = z(23, 12345), z(2, 123), z(13, 12345), z(1, 123), z(12, 12345).

Example 9.14. We can also compute the critical pairs and their corresponding signatures of theappropriate skeleta of B1, . . . , B7:

family critical pair signature(B7)0 (5,6) 5(B6)0 (5,6) 5(B5)0 (5,6) 5(B4)1 (34,45) 3

(35,36) 35(B3)1 (25,26) 25

(35,36) 35(35,45) 3

(B2)2 (235,236) 235(235,245) 23

(B1)2 (125,126) 25(135,136) 35(135,145) 3(235,236) 235(235,245) 23

10. Return to spanning trees of shifted complexes

[Note: We’ll need a better reference to the section on shifted complexes, so we know what N ,etc., mean.]

By Theorem 9.9, then,

detN =

∏(S,T )∈ψ(∆d)

(X1,p + z(S, T ))

Xm1,p

where m is the number of zero eigenvalues of Lud∆,d−1. Since Lud

∆,d−1 acts on ∆d−1, but has ψ(∆d)non-zero eigenvalues (including multiplicity), m = |∆d−1| − |ψ(∆d)| = |Λd−1| − |ψ(∆d−1)|, since∆d−1 = Λd−1 by equation (27) or at least something similar.

Thus,

hd(Σ) = X|Λd−1|1,p

∏σ∈Λd−1

(↑Xσ)

X−|ψ(∆d)|1,p

∏(S,T )∈ψ(∆d)

X1,p + z(S, T )

=

∏σ∈Λd−1

(X1,p ↑Xσ)∏

(S,T )∈ψ(∆d)

X1,p + z(S, T )X1,p

.


By [updated] equation (14a), X1,p ↑Xσ = Xσ∪p. Also,

X1,p + z(S, T )X1,p

=1

X1,p

(X1,p +

∑j∈T XS∪j

↑XS

)=(X1,p ↑XS +

∑j∈T XS∪j

X1,p ↑XS

)=XS∪p +

∑j∈T XS∪j

XS∪p

=

∑j∈T∪pXS∪j

XS∪p.

We finally get, as the first form for hd(Σ),

hd(Σ) =

∏σ∈Λd−1

Xσ

∏(S,T )∈ψ(∆d)

∑j∈T XS∪j

XS

. (40)

There are several things we can do to equation (40) that may or may not make it “better”. It’snot clear which form, if any is best. [Note: I’m not suggesting that all these suggestions shouldstay in the final form of the paper. Whichever ones do will need to be fleshed out better. And, ofcourse, there are probably other things we could do I haven’t listed here.]

• The terms of the second product in equation (40) can be w(S, T , p), where

w(S, T, U) :=

∑j∈T XS∪j∑k∈U XS∪k

.

• Some “tilde”s could be expanded (especially XS in the second product).• Use

∏σ∈Λd−1

Xσ =∏τ∈(starp Σ)d

Xτ , where star is (surprise) the star. Similarly, it may bepossible, in the second product, to use some complex related to ∆ instead of ∆ itself, andsimplify somewhat; but I haven’t found anything good.• Use ∏

(S,T )∈ψ(∆d)

∑j∈T XS∪j

XS

=∏

(S,T )∈ψ(∆d)

1X1,p

∑j∈T XS∪j

↑XS=

∏(S,T )∈ψ(∆d)

z(S, T )X1,p

(41)

• Instead of spreading out the factor Xm1,p into the two products, and making the ↑’s disappear,

we could leave Xm1,p as a separate term. But I like how those seemingly extra factors of X1,p

exactly cancel the ↑’s; it just seems to much of a coincidence to ignore. Also, I don’t know ofan elegant formulation of m, the number of factors of X1,p.• I think we might hope that the denominators of the second product in equation (40) would all

be cancelled by some term in the first product. Certainly, each S that appears is containedin a face of ∆d, and so it is not hard to see that each XS is a factor of some Xσ. It’s notclear to me, though, why there should not be any overlaps here (two different XS ’s tryingto cancel the same Xσ). Also, the resulting answer might be harder to describe in spite ofthe lack of a denominator.• And one tantalizing near-miss, which I include in case you find a way to save it. If only

we were looking at sdu∆,d instead of sud∆,d−1! Here’s why: Of course sdu∆,d and sud∆,d−1 are =-equivalent, so they only differ in their 0’s. But the 0’s of sdu∆,d are counted by homology βd(∆),since this is the top dimension of ∆. And homology just counts faces of ∆ not containing theinitial vertex of ∆ (since ∆ is the deletion, that initial vertex is p+ 1), and not able to addin that vertex. Now bring in 0’s as in Carly’s obstructions to shiftedness paper to combinethe componentwise partial orders of different dimensions. Then not containing the initialvertex, but not being able to add it in, corresponds exactly to a critical pair with signature


0. (Example: If 1 is initial vertex, and 235 is a homology facet, then (0235, 1235) is acritical pair). So if were looking at sdu∆,d instead, then the extra 0 eigenvalues would just bemore critical pairs.

Even though this doesn’t work out, I can’t help but wonder if there is some other way tocorrectly interpret the 0’s of sud∆,d−1 as signatures of critical pairs.

It is not to hard to see that the “coarsening” of equation (40) can be computed something likethis: The left product becomes ∏

σ∈Λd−1

Xσ

=

∏τ∈(starp Σ)d

Xτ

=(Xdeg(starp Σ)d

).

The right product becomes (now let’s just simplify to the case where p = 1)

∏(S,T )∈ψ(∆d)

X〈t〉

X1=∏t

(X〈t〉

X1

)deg∆d(t)−deg∆d

(t+1)

=∏i

(X〈i〉

X1

)(deg∆d)′i

[Note: This isn’t quite right, but something close to it is. I’m having a hard time tracking throughall the partitions and their conjugates, and the change in initial vertex between ∆ and Σ.–Art] Asin Corollary 9.12, the multiplicity means X〈t〉/X1 occurs with multiplicity given by the transposedegree sequence of ∆d. But the extra X1 in the sum (using T instead of T ) throws things off by some,so we just cone ∆ to toss in those extra X1 terms to the summation. See the following example.

Example 10.1. In the bipyramid, (starp Σ)d is the shifted family generated by 135, with faces 123,124, 125, 134, 135. Its degree sequence is 53322. The corresponding left product is

X123X124X125X134X135 = X51X

32X

33X

24X

25 .

Again in the bipyramid, (p ∗ delp Σ)d+1 is the shifted family with two facets, 1234, 1235. Itsdegree sequence is 22211, and the conjugate of the degree sequence is 53. The corresponding rightproduct is (X〈5〉/X1)(X〈3〉/X1)

11. Corollaries

These can be scattered through the paper into appropriate places later. For now, I’m includingthem so we have everything in one file.

11.1. Fine enumerator from shifted to threshold. Let µ be the partition of ”left half (nu-merical) eigenvalues” and λ be the partition of ”right half (numerical) eigenvalues”. Namely, λ isthe partition of eigenvalues not canceled in the alternating product and µ consists of the remainingeigenvalues of the top Laplacian. Write l(µ) for the length of µ.

Let S = S1 > S2 > . . . > SmRevlex be the facets of lk(1), the link of vertex 1, given in reverselexicographic order.

h′′ =l(µ)∏i=1

zSi+l(λ);1

l(λ)∏i

zSi;λi (42)

Consider the one-dimensional case, a threshold graph. In this case, µ consists only of n and λ isthe usual conjugate degree sequence. The link of vertex 1 is simply the neighborhood of 1 which is2, . . . , n for a threshold graph. Hence (S1, . . . , Sn−1) = (2, . . . , n).


Now (42) becomes:

zn;1

n−2∏i=1

zi+1; dTi

=1∑i=1

x1,n

n−1∏i=2

(dTi∑r=1

xmin(i,r),1xmax(i,r),2)

= x1,1xn,2

n−1∏i=2

(dTi∑r=1


which is exactly Theorem 4 from Martin/Reiner.Note: Yes, it is possible to get this with the formulation in equation (40), instead of left and

right eigenvalues, but I’ll have to add that in later. –Art]

11.2. Fine enumerator from threshold to Ferrers. Assume we are given a connected thresholdgraph G on n vertices. We know that G may be partitioned into a clique and an independent setsuch that the neighborhoods of the vertices in the independent set are nested. The vertices of theclique will be the smallest m vertices for some m. Remove all edges (i, j) of G such that i and j ≤ m.The resulting graph will be a Ferrer’s graph. Furthermore all Ferrer’s graphs may be obtained inthis way.

If we begin with the weighted enumerator for threshold graphs and set the terms correspondingto edges (i, j) of G such that i and j ≤ m equal to 0, we can recover the weighted enumerator forFerrer’s graphs.

For a threshold graph we have:

x1,1xn,2

n−1∏i=2

(dTi (G)∑r=1

xmin(i,r),1xmax(i,r),2) (43)

First we break up the expression around the parameter m.

(43) = x1,1xn,2

m∏i=2

(m∑r=1

xmin(i,r),1xmax(i,r),2+dTi (G)∑r=m+1

xmin(i,r),1xmax(i,r),2)n−1∏

i=m+1

(dTi (G)∑r=1


Note that this expression is well defined because if i ≤ m then dTi (G) ≥ m. If i ≤ m and r ≤ mthen max(i, r) ≤ m and these are exactly the terms we wish to set to zero. Hence for Ferrer’s graphswe have:

x1,1xn,2

m∏i=2

(dTi (G)∑r=m+1

xi,1xr,2)n−1∏

i=m+1

(dTi (G)∑r=1

xmin(i,r),1xmax(i,r),2) (44)

For i ≥ m+ 1, we have dTi (G) < m < i so for r ≤ dTi (G), r < i.

(44) = x1,1xn,2

m∏i=2

xi,1

m∏i=2

(dTi (G)∑r=m+1

xr,2)n−1∏

i=m+1

xi,2

n−1∏i=m+1

(dTi (G)∑r=1

xr,1)

= (x1,1x2,1x3,1 . . . xm,1)(xm+1,2xm+2,2 . . . xn,2)m∏i=2

(dTi (G)∑r=m+1

xr,2)n−1∏

i=m+1

(dTi (G)∑r=1

xr,1)


Next we relate the degree sequences of G and F . Let F1 be the first m vertices of F . Then wehave:

(dTm+1(G) . . . dTn−1(G)) = (dT2 (F1) . . . dTn−m(F1)).

The only contribution to the left hand side comes from vertices in G which have label less than orequal to m, i.e. vertices in the clique. (No vertex of label greater than m can have degree greaterthan m.) But when we move from G to F we exactly reduce the degree of each of these vertices bym− 1.

Let F2 be the second n−m vertices of F . In this case we have:

(dT2 (G) . . . dTm(G)) = (dT2 (F2) +m. . . dTm(F2) +m).

This is due to the fact that the degrees of the vertices of F2 do not change in moving from G to F .Finally, make a change of variables sending xr,1 to xr and xr,2 to yr−m. From this perspective

we have the vertices of F1 labeled x1, . . . xm and the vertices of F2 labeled y1, . . . yn−mTherefore we have:

(x1 . . . xm)(y1 . . . yn−m)m∏i=2

(dTi (F2)∑r=1

yr)n−m∏i=2

(dTi (F1)∑r=1

xr) (45)

which is Theorem 2.1 of Ehrenborg/van Willigenburg.

References

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[2] Eric Babson and Isabella Novik, Face numbers and nongeneric initial ideals, Electron. J. Combin. 11 (2004/06),

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Adv. Stud. Pure Math., 33, Math. Soc. Japan, Tokyo, 2002.[17] J.W. Moon, Counting Labeled Trees. Canadian Mathematical Monographs, No. 1, Canadian Mathematical Con-

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Department of Mathematical Sciences, University of Texas at El Paso

Departments of Mathematics and Computer Science, The University of Chicago

Department of Mathematics, University of Kansas

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