mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a ranked

partially ordered set 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 ...

or ranked poset may be either: * 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 ...

, or * a poset with the property that for every element ''x'', all maximal

chains A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A c ...

among those with ''x'' as

greatest element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an eleme ...

have the same finite

length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...

, or * a poset in which all maximal chains have the same finite length. The second definition differs from the first in that it requires all minimal elements to have the same rank; for posets with a least element, however, the two requirements are equivalent. The third definition is even more strict in that it excludes posets with infinite chains and also requires all maximal elements to have the same rank.

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 ...

defines a graded poset of length ''n'' as one in which all maximal chains have length ''n''.Richard Stanley, ''Enumerative Combinatorics,'' vol.1 p.99, Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, 1995,

References

Order theory {{combin-stub