Let P be a d-dimensional simplicial polytope. For i = 0, 1, ..., d − 1, let fi denote the number of i-dimensional faces of P. The sequence
![{\displaystyle f(P)=(f_{0},f_{1},\ldots ,f_{d-1})}](//wikimedia.org/api/rest_v1/media/math/render/svg/ebe43d9a728beb9bcd0eda1473cc1c0fee6fd53e)
is called the f-vector of the polytope P. Additionally, set
![{\displaystyle f_{-1}=1,f_{d}=1.}](//wikimedia.org/api/rest_v1/media/math/render/svg/210780795fae827949b40c1a9e7fdb1fff87353d)
Then for any k = −1, 0, ..., d − 2, the following Dehn–Sommerville equation holds:
![{\displaystyle \sum _{j=k}^{d-1}(-1)^{j}{\binom {j+1}{k+1}}f_{j}=(-1)^{d-1}f_{k}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/ce7aa3387f3677652ecae985c41a2cd02a124314)
When k = −1, it expresses the fact that Euler characteristic of a (d − 1)-dimensional simplicial sphere is equal to 1 + (−1)d − 1.
Dehn–Sommerville equations with different k are not independent. There are several ways to choose a maximal independent subset consisting of
equations. If d is even then the equations with k = 0, 2, 4, ..., d − 2 are independent. Another independent set consists of the equations with k = −1, 1, 3, ..., d − 3. If d is odd then the equations with k = −1, 1, 3, ..., d − 2 form one independent set and the equations with k = −1, 0, 2, 4, ..., d − 3 form another.
Sommerville found a different way to state these equations:
![{\displaystyle \sum _{i=-1}^{k-1}(-1)^{d+i}{\binom {d-i-1}{d-k}}f_{i}=\sum _{i=-1}^{d-k-1}(-1)^{i}{\binom {d-i-1}{k}}f_{i},}](//wikimedia.org/api/rest_v1/media/math/render/svg/e61c3b4f94b01c75916acc21162d408e536117ad)
where 0 ≤ k ≤ 1⁄2(d−1). This can be further facilitated introducing the notion of h-vector of P. For k = 0, 1, ..., d, let
![{\displaystyle h_{k}=\sum _{i=0}^{k}(-1)^{k-i}{\binom {d-i}{k-i}}f_{i-1}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/81c0bacfd0dd68e7087aa28ed2b8c9ef4e76c5c1)
The sequence
![{\displaystyle h(P)=(h_{0},h_{1},\ldots ,h_{d})}](//wikimedia.org/api/rest_v1/media/math/render/svg/6b18866960006c568ae4c2f177963025db18f8fa)
is called the h-vector of P. The f-vector and the h-vector uniquely determine each other through the relation
![{\displaystyle \sum _{i=0}^{d}f_{i-1}(t-1)^{d-i}=\sum _{k=0}^{d}h_{k}t^{d-k}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/7b1becf84582b2b93a1cee0c53df9f41be575a38)
Then the Dehn–Sommerville equations can be restated simply as
![{\displaystyle h_{k}=h_{d-k}\quad {\text{ for }}0\leq k\leq d.}](//wikimedia.org/api/rest_v1/media/math/render/svg/dacd406a83cbd382086f255d7ef6f9bc8073fc2c)
The equations with 0 ≤ k ≤ 1⁄2(d−1) are independent, and the others are manifestly equivalent to them.
Richard Stanley gave an interpretation of the components of the h-vector of a simplicial convex polytope P in terms of the projective toric variety X associated with (the dual of) P. Namely, they are the dimensions of the even intersection cohomology groups of X:
![{\displaystyle h_{k}=\dim _{\mathbb {Q} }\operatorname {IH} ^{2k}(X,\mathbb {Q} )}](//wikimedia.org/api/rest_v1/media/math/render/svg/0762c448ff24a0d554063d9eea32101a7a69d8be)
(the odd intersection cohomology groups of X are all zero). In this language, the last form of the Dehn–Sommerville equations, the symmetry of the h-vector, is a manifestation of the Poincaré duality in the intersection cohomology of X.