Order polynomial explained
The order polynomial is a polynomial studied in mathematics, in particular in algebraic graph theory and algebraic combinatorics. The order polynomial counts the number of order-preserving maps from a poset to a chain of length
. These order-preserving maps were first introduced by
Richard P. Stanley while studying ordered structures and partitions as a Ph.D. student at
Harvard University in 1971 under the guidance of
Gian-Carlo Rota.
Definition
Let
be a
finite poset with
elements denoted
, and let
be a
chain
elements. A map
is order-preserving if
implies
. The number of such maps grows polynomially with
, and the function that counts their number is the
order polynomial
.
Similarly, we can define an order polynomial that counts the number of strictly order-preserving maps
, meaning
implies
. The number of such maps is the
strict order polynomial \Omega\circ(n)=\Omega\circ(P,n)
.
[1] Both
and
have degree
. The order-preserving maps generalize the
linear extensions of
, the order-preserving bijections
\phi:P\stackrel{\sim}{\longrightarrow}[p]
. In fact, the leading coefficient of
and
is the number of linear extensions divided by
.
Examples
Letting
be a
chain of
elements, we have
\Omega(n)=\binom{n+p-1}{p}=\left(\left({n\atopp}\right)\right)
and \Omega\circ(n)=\binom{n}{p}.
There is only one linear extension (the identity mapping), and both polynomials have leading term
.
Letting
be an
antichain of
incomparable elements, we have
\Omega(n)=\Omega\circ(n)=np
. Since any bijection
\phi:P\stackrel{\sim}{\longrightarrow}[p]
is (strictly) order-preserving, there are
linear extensions, and both polynomials reduce to the leading term
.
Reciprocity theorem
There is a relation between strictly order-preserving maps and order-preserving maps:[2]
\Omega\circ(n)=(-1)|P|\Omega(-n).
In the case that
is a chain, this recovers the negative binomial identity. There are similar results for the
chromatic polynomial and
Ehrhart polynomial (see below), all special cases of Stanley's general
Reciprocity Theorem.
[3] Connections with other counting polynomials
Chromatic polynomial
counts the number of proper
colorings of a finite
graph
with
available colors. For an
acyclic orientation
of the edges of
, there is a natural "downstream" partial order on the vertices
implied by the basic relations
whenever
is a directed edge of
. (Thus, the
Hasse diagram of the poset is a subgraph of the oriented graph
.) We say
is compatible with
if
is order-preserving. Then we have
P(G,n) = \sum\sigma\Omega\circ(\sigma,n),
where
runs over all acyclic orientations of
G, considered as poset structures.
[4] Order polytope and Ehrhart polynomial
See main article: Order polytope. The order polytope associates a polytope with a partial order. For a poset
with
elements, the order polytope
is the set of order-preserving maps
, where
[0,1]=\{t\inR\mid0\leqt\leq1\}
is the ordered unit interval, a continuous chain poset.
[5] [6] More geometrically, we may list the elements
, and identify any mapping
with the point
(f(x1),\ldots,f(xp))\inRp
; then the order polytope is the set of points
with
if
.
[7] The Ehrhart polynomial counts the number of integer lattice points inside the dilations of a polytope. Specifically, consider the lattice
and a
-dimensional polytope
with vertices in
; then we define
the number of lattice points in
, the dilation of
by a positive integer scalar
.
Ehrhart showed that this is a
rational polynomial of degree
in the variable
, provided
has vertices in the lattice.
[8] In fact, the Ehrhart polynomial of an order polytope is equal to the order polynomial of the original poset (with a shifted argument):[9]
L(O(P),n) = \Omega(P,n{+}1).
This is an immediate consequence of the definitions, considering the embedding of the
-chain poset
[n{+}1]=\{0<1< … <n\}\subsetR
.
Notes and References
- Book: Stanley. Richard P.. Ordered structures and partitions. 1972. American Mathematical Society. Providence, Rhode Island.
- Stanley . Richard P. . A chromatic-like polynomial for ordered sets . 1970 . Proc. Second Chapel Hill Conference on Combinatorial Mathematics and Its Appl. . 421 - 427.
- Book: Enumerative combinatorics. Volume 1. Stanley . Richard P.. 2012. Cambridge University Press. 9781139206549. 2nd. New York. 4.5.14 Reciprocity theorem for linear homogeneous diophantine equations. 777400915.
- Stanley. Richard P.. Acyclic orientations of graphs. Discrete Mathematics. 1973. 5. 2. 171–178. 10.1016/0012-365X(73)90108-8.
- Alexander . Karzanov . Leonid . Khachiyan . On the conductance of Order Markov Chains . . 8 . 7 - 15 . 1991 . 10.1007/BF00385809. 120532896 .
- Graham . Brightwell . Peter . Winkler . Counting linear extensions . . 8 . 225 - 242 . 1991 . 3 . 10.1007/BF00383444. 119697949 .
- Stanley. Richard P.. 1986. Two poset polytopes. Discrete & Computational Geometry. 1. 9–23. 10.1007/BF02187680. free.
- Book: Computing the continuous discretely. Computing the Continuous Discretely . Beck. Matthias. Robins. Sinai. Springer. 2015. 978-1-4939-2968-9. New York. 64–72.
- Linial. Nathan. 1984. The information-theoretic bound is good for merging. SIAM J. Comput.. 13. 4. 795 - 801. 10.1137/0213049.
Kahn . Jeff . Kim . Jeong Han . 10.1006/jcss.1995.1077 . 3 . Journal of Computer and System Sciences . 390–399 . Entropy and sorting. . 51 . 1995. free .