# H-vector

An **h-vector** is a sequence of integers \( (h_0, h_1, \ldots, h_d) \) attached to a finite simplicial complex (or, in one of several variants, to a polytope or poset) that encodes the number of faces the complex has in each dimension.<sup>[1](https://www.sciencedirect.com/science/article/abs/pii/S0196885822001609)</sup> It is a linear transform of the more direct f-vector, the list of face counts, and it is preferred in many settings because it absorbs the binomial incidence data of a simplex and exposes linear relations among face numbers that are invisible in the f-vector itself.<sup>[2](https://androma.org/page/Geometric%20Combinatorics)</sup> Since its introduction, the h-vector has been described as perhaps the most widely studied combinatorial invariant of a simplicial complex, and its properties continue to drive research.<sup>[1](https://www.sciencedirect.com/science/article/abs/pii/S0196885822001609)</sup>

| Key fact | Detail |
|---|---|
| What it encodes | One way of encoding the number of faces of a finite simplicial complex in each dimension.<sup>[1](https://www.sciencedirect.com/science/article/abs/pii/S0196885822001609)</sup> |
| Relation to the f-vector | The f-vector and h-vector determine each other; for a complex of dimension r the h-vector is read off from the Hilbert series \( \mathrm{HS}(S/I_{\Delta}; t) = (h_0 + h_1 t + \cdots + h_{r+1} t^{r+1})/(1-t)^{r+1} \) of the Stanley–Reisner ring.<sup>[3](https://doi.org/10.1016/j.aim.2014.11.002)</sup> |
| Shelling meaning | For simplicial polytopes the h-numbers have a natural interpretation in terms of shellings, via the "obligatory" vertices of each new facet.<sup>[4](https://www.researchgate.net/profile/Fabrice-Clerot/post/Could_anyone_tell_me_what_the_mathematical_definition_expression_for_a_polyhedral_convex_set_and_polyhedral_cone_in_general_Euclidean_space_is/attachment/59d61db679197b80779797ea/AS%3A272440477847554%401441966280124/download/Gallier%2B-%2BConvex%2Bsets%2Betc.pdf)</sup> |
| Dehn–Sommerville symmetry | For a simplicial d-polytope the h-vector satisfies \( h_i = h_{d-i} \), so the second half of the vector is determined by the first.<sup>[4](https://www.researchgate.net/profile/Fabrice-Clerot/post/Could_anyone_tell_me_what_the_mathematical_definition_expression_for_a_polyhedral_convex_set_and_polyhedral_cone_in_general_Euclidean_space_is/attachment/59d61db679197b80779797ea/AS%3A272440477847554%401441966280124/download/Gallier%2B-%2BConvex%2Bsets%2Betc.pdf)</sup> |
| Complete characterization | The g-theorem (Billera–Lee and Stanley, 1980) characterizes exactly which integer vectors arise as h-vectors of simplicial polytopes.<sup>[5](https://citeseerx.ist.psu.edu/document?doi=709ae0ef7a0137b548817e3c4d61a113b8b390dd&repid=rep1&type=pdf)</sup> |
| Sign | The h-vector of a shellable complex is nonnegative, but for arbitrary complexes entries can be negative; a filled triangle with two attached edges has h-vector \( (1, 2, -2, 0) \).<sup>[6](https://www.ms.uky.edu/~Lee/ma714fa13/notes.pdf)</sup><sup> • </sup><sup>[7](https://macaulay2.com/doc/OLD/Macaulay2-1.22/share/doc/Macaulay2/SimplicialDecomposability/html/_h__Vector.html)</sup>  |

## Definition and relation to the f-vector

For a simplicial complex K of dimension n − 1, let \( f_j \) denote the number of j-dimensional faces, with \( f_{-1} = 1 \) counting the empty face. The h-numbers \( h_0, \ldots, h_n \) are defined by the polynomial identity

\[ \sum_{j=0}^{n} h_j t^{n-j} = \sum_{j=-1}^{n-1} f_j (t-1)^{n-1-j}, \]

and \( h(K) = (h_0, \ldots, h_n) \) is the h-vector of K.<sup>[8](https://eprints.maths.manchester.ac.uk/1055/1/tortoptet6.pdf)</sup> The transform is invertible, so the f-vector and the h-vector determine each other.<sup>[9](https://doi.org/10.1016/j.jalgebra.2019.12.020)</sup> Its purpose is to repackage face counts so that the underlying linear relations become explicit; the h-vector maintains the combinatorial data of the f-vector while making those relations visible.<sup>[10](https://www.math.uwaterloo.ca/~hwolkowi/patrickabstract.html)</sup>

There is also a direct combinatorial description for shellable complexes. In a shelling \( F_1, \ldots, F_s \) of a d-dimensional complex, the restriction \( R_j \) of facet \( F_j \) is the set of "obligatory" vertices, those \( v \in F_j \) for which \( F_j \setminus \{v\} \) is contained in some earlier facet; the h-numbers count facets by the size of their restriction sets.<sup>[4](https://www.researchgate.net/profile/Fabrice-Clerot/post/Could_anyone_tell_me_what_the_mathematical_definition_expression_for_a_polyhedral_convex_set_and_polyhedral_cone_in_general_Euclidean_space_is/attachment/59d61db679197b80779797ea/AS%3A272440477847554%401441966280124/download/Gallier%2B-%2BConvex%2Bsets%2Betc.pdf)</sup>

## The algebraic formulation

The most powerful tool for studying h-vectors is commutative algebra. For a complex Δ on vertices \( x_1, \ldots, x_n \) over a field k, the Stanley–Reisner ring (or face ring) is \( k[\Delta] = k[x_1, \ldots, x_n]/I_{\Delta} \), where \( I_{\Delta} \) is generated by the monomials whose supports are not faces of Δ.<sup>[6](https://www.ms.uky.edu/~Lee/ma714fa13/notes.pdf)</sup> The h-vector appears as the coefficients of the Hilbert series of this ring modulo a linear system of parameters: if \( B = k[\Delta]/(\theta_1, \ldots, \theta_d) \), then \( \sum_i \dim_k B_i \, t^i = h(\Delta, t) \), so \( \dim_k B_i = h_i \).<sup>[6](https://www.ms.uky.edu/~Lee/ma714fa13/notes.pdf)</sup><sup> • </sup><sup>[3](https://doi.org/10.1016/j.aim.2014.11.002)</sup>

This algebraic viewpoint explains the sign behavior. Stanley proved that if Δ is shellable, then suitable parameters exist, the ring is Cohen–Macaulay, and the h-vector is an M-vector, meaning it is the Hilbert function of a standard graded algebra and hence nonnegative.<sup>[6](https://www.ms.uky.edu/~Lee/ma714fa13/notes.pdf)</sup> Reisner's theorem characterizes Cohen–Macaulay complexes homologically, and simplicial balls and spheres are Cohen–Macaulay whether or not they are shellable.<sup>[6](https://www.ms.uky.edu/~Lee/ma714fa13/notes.pdf)</sup> Outside this class the guarantee disappears: the [Macaulay2](https://www.edgepedia.org/macaulay2) documentation computes an h-vector of \( (1, 2, -2, 0) \) for a small nonpure complex, showing that arbitrary complexes can have negative entries.<sup>[7](https://macaulay2.com/doc/OLD/Macaulay2-1.22/share/doc/Macaulay2/SimplicialDecomposability/html/_h__Vector.html)</sup> Nonpure sequentially Cohen–Macaulay complexes can likewise have negative entries.<sup>[11](https://www.sk.tsukuba.ac.jp/~hachi/archives/hereditary.pdf)</sup>

## Dehn–Sommerville relations

For simplicial polytopes the h-vector satisfies linear relations beyond the definition. These equations were found by Dehn for dimensions 4 and 5 in 1905 and in general by Sommerville in 1927, and in h-vector form they read simply \( h_i = h_{d-i} \); the proof is short in the language of shellings.<sup>[4](https://www.researchgate.net/profile/Fabrice-Clerot/post/Could_anyone_tell_me_what_the_mathematical_definition_expression_for_a_polyhedral_convex_set_and_polyhedral_cone_in_general_Euclidean_space_is/attachment/59d61db679197b80779797ea/AS%3A272440477847554%401441966280124/download/Gallier%2B-%2BConvex%2Bsets%2Betc.pdf)</sup> The classical statement is that the h-vector of an Eulerian simplicial complex is symmetric.<sup>[12](https://arxiv.org/abs/2003.00160)</sup> The symmetry means that everything about the face numbers of a simplicial polytope is determined by the first half of the h-vector.<sup>[13](https://arxiv.org/abs/1906.05859)</sup>

The relations extend beyond the Eulerian case. For an arbitrary pure \( (d-1) \)-dimensional complex, the difference \( h_{d-i}(\Delta) - h_i(\Delta) \) can be expressed in terms of the Euler characteristics of links of faces, and analogous formulas hold for flag h-vectors of balanced complexes and for toric h-vectors of graded posets, extending Klee's semi-Eulerian relations.<sup>[12](https://arxiv.org/abs/2003.00160)</sup> Bayer and Billera generalized the Dehn–Sommerville relations to the flag h-vector of polytopes, spheres, and Eulerian posets in 1985.<sup>[14](https://citeseerx.ist.psu.edu/document?doi=7c00981c042ddde4e799658090d5f9261d63d282&repid=rep1&type=pdf)</sup>

## The g-theorem and the g-conjecture

The central characterization question asks which integer vectors occur as h-vectors of simplicial polytopes. McMullen conjectured an answer in 1971 in terms of the g-vector, defined by \( g_0 = 1 \) and \( g_i = h_i - h_{i-1} \) for \( 1 \le i \le \lfloor d/2 \rfloor \): the [Dehn–Sommerville equations](https://www.edgepedia.org/dehn-sommerville-equations) should hold and the g-vector should be an M-sequence.<sup>[8](https://eprints.maths.manchester.ac.uk/1055/1/tortoptet6.pdf)</sup><sup> • </sup><sup>[13](https://arxiv.org/abs/1906.05859)</sup>  The result, now called the g-theorem, was completed in 1980 and 1981: Billera and Lee constructed polytopes establishing sufficiency of McMullen's conditions, and Stanley proved necessity.<sup>[5](https://citeseerx.ist.psu.edu/document?doi=709ae0ef7a0137b548817e3c4d61a113b8b390dd&repid=rep1&type=pdf)</sup> The g-vector is in a sense a minimal encoding of the f-vector of a simplicial polytope.<sup>[14](https://citeseerx.ist.psu.edu/document?doi=7c00981c042ddde4e799658090d5f9261d63d282&repid=rep1&type=pdf)</sup>

Stanley's route ran through algebra. For his 1975 proof of the Upper Bound Conjecture, he needed a ring whose Hilbert function relates to the f-vector of a triangulated sphere, and found it in the Stanley–Reisner ring; the proof used Reisner's theorem that this ring is Cohen–Macaulay.<sup>[15](https://onlinelibrary.wiley.com/doi/pdf/10.1002/sapm1975542135)</sup><sup> • </sup><sup>[16](https://dspace.mit.edu/entities/publication/d390a619-3973-4397-b14c-e022a0346536)</sup>  In 1980 he observed that McMullen's conjecture would follow from a hard Lefschetz property of the face ring, which he proved for polytope spheres using algebraic geometry.<sup>[13](https://arxiv.org/abs/1906.05859)</sup> The general case for arbitrary simplicial spheres remained open until Karim Adiprasito, a combinatorialist then working on combinatorial [Hodge theory](https://www.edgepedia.org/hodge-theory), resolved the g-conjecture in full generality in 2018, characterizing the face numbers of simplicial rational homology spheres with a version of the Kähler package that works beyond positivity.<sup>[17](https://arxiv.org/abs/1812.10454)</sup>

## Extremal examples and inequalities

The h-vector controls extremal face counting. For a simplicial d-polytope with n vertices, the cyclic polytopes achieve

\[ h_i = \binom{n-d-1+i}{i} \quad \text{for } 0 \le i \le \lfloor d/2 \rfloor, \]

with the remaining entries fixed by Dehn–Sommerville symmetry; this is the pattern behind McMullen's Upper Bound Theorem, under which cyclic polytopes have the largest possible face numbers among simplicial d-polytopes with n vertices.<sup>[2](https://androma.org/page/Geometric%20Combinatorics)</sup> At the other end, the Generalized Lower Bound Conjecture of McMullen and Walkup asserts unimodality of the h-vector of simplicial polytopes.<sup>[14](https://citeseerx.ist.psu.edu/document?doi=7c00981c042ddde4e799658090d5f9261d63d282&repid=rep1&type=pdf)</sup>

Inequality questions extend to other classes. The h-vector of the independence complex of a matroid was proved log-concave, \( h_i^2 \ge h_{i-1} h_{i+1} \), independently by Ardila, Denham, and Huh and by Berget, Spink, and Tseng in 2023; a related conjecture of Brylawski on ultra log-concavity of broken circuit complex h-vectors has small counterexamples, such as a rank-4 matroid on 6 elements whose independence h-vector is \( (1, 2, 2, 2, 2) \).<sup>[18](https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/complementary-vectors-of-simplicial-complexes/27A13C257BF97E542E21509201273176)</sup> A long-standing conjecture of Stanley asserts that the h-vector of a matroid independence complex is a pure O-sequence.<sup>[18](https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/complementary-vectors-of-simplicial-complexes/27A13C257BF97E542E21509201273176)</sup>

Monotonicity gives another family of inequalities: if Δ is a Cohen–Macaulay complex of dimension d − 1 and Δ′ is a Cohen–Macaulay subcomplex of the same dimension, then \( h(\Delta') \le h(\Delta) \) entrywise, a result obtained earlier in a different form by [Gil Kalai](https://www.edgepedia.org/gil-kalai); the same techniques yield an analogous monotonicity for Ehrhart polynomials of lattice polytopes.<sup>[19](https://www.sciencedirect.com/science/article/pii/S0195669883710280)</sup>

## Variants

Several modified h-vectors serve classes of objects beyond simplicial complexes. The toric h-vector applies to a significantly larger class of objects than simplicial polytopes and makes explicit the relationship between polytope theory and algebraic geometry; for a simplicial polytope it agrees with the ordinary h-vector, whose entries appear as the (even) Betti numbers of the associated toric variety.<sup>[10](https://www.math.uwaterloo.ca/~hwolkowi/patrickabstract.html)</sup><sup> • </sup><sup>[20](https://www.math.ucdavis.edu/~deloera/TEACHING/RMMC2011/alggeopoly.pdf)</sup>  The flag h-vector, indexed by subsets of ranks, records chains of faces and satisfies the Bayer–Billera generalization of Dehn–Sommerville.<sup>[14](https://citeseerx.ist.psu.edu/document?doi=7c00981c042ddde4e799658090d5f9261d63d282&repid=rep1&type=pdf)</sup> For lattice polytopes, the Ehrhart h*-vector plays the analogous role for counting lattice points in dilates: the Ehrhart h*-vector of an integer Gorenstein polytope with a regular unimodular triangulation satisfies the g-theorem and is in particular unimodal.<sup>[21](https://www.arxiv.symmetricfunctions.com/paper/math/0508392v2)</sup> Cubical complexes admit an h-vector computed through plane trees for shellable complexes.<sup>[22](https://epubs.siam.org/doi/abs/10.1137/0404049)</sup>

## References

1. [Algebraic h-vectors of simplicial complexes through local cohomology, part 1, Advances in Applied Mathematics 145 (2023)](https://www.sciencedirect.com/science/article/abs/pii/S0196885822001609)
2. [Geometric Combinatorics, Androma reference page](https://androma.org/page/Geometric%20Combinatorics)
3. [h-Vectors of matroids and logarithmic concavity, Advances in Mathematics (2015)](https://doi.org/10.1016/j.aim.2014.11.002)
4. [J. Gallier, Convex Sets, etc., Chapter 7: Dehn–Sommerville equations for simplicial polytopes](https://www.researchgate.net/profile/Fabrice-Clerot/post/Could_anyone_tell_me_what_the_mathematical_definition_expression_for_a_polyhedral_convex_set_and_polyhedral_cone_in_general_Euclidean_space_is/attachment/59d61db679197b80779797ea/AS%3A272440477847554%401441966280124/download/Gallier%2B-%2BConvex%2Bsets%2Betc.pdf)
5. [The g-Theorem, CiteSeerX course notes](https://citeseerx.ist.psu.edu/document?doi=709ae0ef7a0137b548817e3c4d61a113b8b390dd&repid=rep1&type=pdf)
6. [Polytopes Course Notes, University of Kentucky (Lee)](https://www.ms.uky.edu/~Lee/ma714fa13/notes.pdf)
7. [hVector — the h-vector of a simplicial complex, Macaulay2 documentation](https://macaulay2.com/doc/OLD/Macaulay2-1.22/share/doc/Macaulay2/SimplicialDecomposability/html/_h__Vector.html)
8. [An Invitation to Toric Topology, University of Manchester preprint](https://eprints.maths.manchester.ac.uk/1055/1/tortoptet6.pdf)
9. [Simplicial complexes and tilting theory for Brauer tree algebras, Journal of Algebra (2020)](https://doi.org/10.1016/j.jalgebra.2019.12.020)
10. [Vector spaces spanned by the angle sums of polytopes, ResearchGate](https://www.math.uwaterloo.ca/~hwolkowi/patrickabstract.html)
11. [Hachimura, Hereditary properties and h-vectors, University of Tsukuba](https://www.sk.tsukuba.ac.jp/~hachi/archives/hereditary.pdf)
12. [Connor Sawaske and Lei Xue, Non-Eulerian Dehn–Sommerville relations, arXiv:2003.00160 (2020)](https://arxiv.org/abs/2003.00160)
13. [Karim Adiprasito, FAQ on the g-theorem and the hard Lefschetz theorem for face rings, arXiv:1906.05859 (2019)](https://arxiv.org/abs/1906.05859)
14. [Generalized lower bound and flag vector notes, CiteSeerX (University of Kentucky)](https://citeseerx.ist.psu.edu/document?doi=7c00981c042ddde4e799658090d5f9261d63d282&repid=rep1&type=pdf)
15. [Richard P. Stanley, The Upper Bound Conjecture and Cohen–Macaulay Rings, Studies in Applied Mathematics 54 (1975), 135–142](https://onlinelibrary.wiley.com/doi/pdf/10.1002/sapm1975542135)
16. [Richard P. Stanley, How the Upper Bound Conjecture Was Proved, Annals of Combinatorics 18 (2014), 533–539, MIT DSpace](https://dspace.mit.edu/entities/publication/d390a619-3973-4397-b14c-e022a0346536)
17. [Karim Adiprasito, Combinatorial Lefschetz theorems beyond positivity, arXiv:1812.10454 (2018)](https://arxiv.org/abs/1812.10454)
18. [Complementary vectors of simplicial complexes, Forum of Mathematics, Sigma 14 (2026), e133](https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/complementary-vectors-of-simplicial-complexes/27A13C257BF97E542E21509201273176)
19. [A Monotonicity Property of h-vectors and h*-vectors, European Journal of Combinatorics 14 (1993), 251–258](https://www.sciencedirect.com/science/article/pii/S0195669883710280)
20. [Polytopes and Commutative Algebra, UC Davis lecture notes (De Loera)](https://www.math.ucdavis.edu/~deloera/TEACHING/RMMC2011/alggeopoly.pdf)
21. [h-vectors of Gorenstein polytopes, arXiv:math/0508392](https://www.arxiv.symmetricfunctions.com/paper/math/0508392v2)
22. [H-Vectors of Shellable Cubical Complexes, SIAM Journal on Discrete Mathematics](https://epubs.siam.org/doi/abs/10.1137/0404049)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Logic and discrete mathematics › General discrete mathematics and discrete structures › Combinatorics › Geometric, polyhedral, and topological combinatorics › Face numbers and face vectors*

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