combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...

mathematics, an Eulerian poset is a

graded poset In mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) ''P'' equipped with a rank function ''ρ'' from ''P'' to the set N of all natural numbers. ''ρ'' must satisfy the following two properties: * Th ...

in which every nontrivial interval has the same number of elements of even rank as of odd rank. An Eulerian poset which is a

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...

is an Eulerian lattice. These objects are named after

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

. Eulerian lattices generalize

face lattice A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

s of

convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

s and much recent research has been devoted to extending known results from

polyhedral combinatorics Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral comb ...

, such as various restrictions on ''f''-vectors of convex

simplicial polytope In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only triangular facesPolyhedra, Peter R. Cromwell, 1997. (p.341) and corresponds via Steinitz's ...

s, to this more general setting.

Examples

* The

of a

, consisting of its faces, together with the smallest element, the empty face, and the largest element, the polytope itself, is an Eulerian lattice. The odd–even condition follows from

Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for an ...

. * Any simplicial generalized homology sphere is an Eulerian lattice. * Let ''L'' be a regular

cell complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...

such that , ''L'', is a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

with the same Euler characteristic as the

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

of the same dimension (this condition is vacuous if the dimension is odd). Then the

poset In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...

of cells of ''L'', ordered by the inclusion of their closures, is Eulerian. * Let ''W'' be a

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...

with

Bruhat order In mathematics, the Bruhat order (also called strong order or strong Bruhat order or Chevalley order or Bruhat–Chevalley order or Chevalley–Bruhat order) is a partial order on the elements of a Coxeter group, that corresponds to the inclusion o ...

. Then (''W'',≤) is an Eulerian poset.

Properties

* The defining condition of an Eulerian poset ''P'' can be equivalently stated in terms of its

Möbius function The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...

: ::

\mu_P(x,y)=(-1)^ \text x\leq y.

* The dual of an Eulerian poset with a top element, obtained by reversing the partial order, is Eulerian. * Richard Stanley defined the toric ''h''-vector of a

ranked poset In mathematics, a ranked partially ordered set or ranked poset may be either: * a graded poset, or * a poset with the property that for every element ''x'', all maximal chains among those with ''x'' as greatest element In mathematics, especia ...

, which generalizes the ''h''-vector of a simplicial polytope. He proved that the

Dehn–Sommerville equations In mathematics, the Dehn–Sommerville equations are a complete set of linear relations between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5, they were found by Max Dehn in 1905. Their gen ...

h_k = h_ \,

: hold for an arbitrary Eulerian poset of rank ''d'' + 1.''Enumerative combinatorics'', Theorem 3.14.9 However, for an Eulerian poset arising from a regular cell complex or a convex polytope, the toric ''h''-vector neither determines, nor is neither determined by the numbers of the cells or faces of different dimension and the toric ''h''-vector does not have a direct combinatorial interpretation.

Notes

References

Richard P. Stanley Richard Peter Stanley (born June 23, 1944) is an Emeritus Professor of Mathematics at the Massachusetts Institute of Technology, in Cambridge, Massachusetts. From 2000 to 2010, he was the Norman Levinson Professor of Applied Mathematics. He r ...

''Enumerative Combinatorics''
Volume 1. Cambridge University Press, 1997

Examples

Properties

Notes

References

See also