Order polynomial

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 $n$ . 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 $P$ be a finite poset, $x$ and $y$ be points in $P$ and $[n]$ be a chain of length $n$ . A map $\phi :P\to [n]$ is order-preserving if $x\leq y$ implies $\phi (x)\leq \phi (y)$ . The number of such maps grows polynomially with $n$ , and the function that counts their number as a function of $n$ is the order polynomial $\Omega (n)$ .

Similarly, we can define an order polynomial that counts the number of strictly order-preserving maps. A map $\phi :P\to [n]$ is strictly order-preserving if $x<y$ implies $\phi (x)<\phi (y)$ . The order polynomial that counts the number of these maps is denoted by $\Omega ^{\circ }(n)$ .^[1]

Examples

Let $P$ be a chain of length $k$ . Then $\Omega (n)={\binom {n+k-1}{k}}$ and $\Omega ^{\circ }(n)={\binom {n}{k}}.$

We can also consider the poset $P$ with $p$ disjoint points. Then the number of order-preserving maps to a chain of length $n$ is $\Omega ^{\circ }(n)=n^{p}$ , and $\Omega ^{\circ }(n)=n^{p}$ .

Reciprocity theorem

The reciprocity theorem shows that there is a relation between strictly order-preserving maps and order-preserving maps. In addition, this comes hand in hand with important properties that the chromatic polynomial and Ehrhart polynomial share. The relation can be stated as follows:^[2]

\Omega ^{\circ }(n)=(-1)^{|P|}\Omega (-n).

In the case that $P$ is a chain, this recovers the negative binomial identity.

Connections

Chromatic polynomial

The chromatic polynomial counts the number of proper ways to color a graph. Let $G$ be a finite graph and let $\sigma$ be an acyclic orientation of $G$ . Then there is a natural binary relation on the vertices $v,u$ of $G$ defined as $u\geq v$ if $u\rightarrow v$ . In particular, $\sigma$ is a partial ordering on the set of vertices $V(G)$ of $G$ . In addition, $\phi :V(G)\rightarrow \lbrace 1,2,3,\ldots ,n\rbrace$ is compatible with $\sigma$ if $\phi$ is order-preserving. Then, we can conclude that

P(G,n)=\sum _{\sigma }\Omega ^{\circ }(n),

where $\sigma$ are all acyclic orientations of G.^[3]

Ehrhart polynomial

The Ehrhart polynomial is a polynomial that associates the number of integer lattice points and the expansion of a polytope $P$ by a factor of $t$ . In other words, consider the lattice $L$ and a $d$ -dimensional polytope in $\mathbb {R} ^{n}$ with integer vertices. Let $t$ be a positive integer, and $tP$ be a dilation of $P$ so

L(P,t)=\#(tP\cap L)

is the number of lattice points in $tP$ . Eugène Ehrhart showed that this was a rational polynomial of degree $d$ in $t$ .^[4]

Order polytope

The order polytope associates a polytope with a partial order. Let $P$ be a poset with $p$ elements. Then the order polytope, denoted $O(P)$ , is the set of points in the $p$ -dimensional unit cube such that the coordinates satisfy the partial order.^[5]^[6]

More formally, let $\mathbb {R} ^{|P|}$ be the set of all functions from $P$ to $\mathbb {R}$ . The order polytope $O(P)$ of the poset $P$ is the set of all maps $\phi$ in $\mathbb {R} ^{|P|}$ which satisfy the following two conditions. First, $0\leq \phi (x)\leq 1$ for all $x\in P$ , and second, $f(x)\leq f(y)$ if $x\leq y$ in the ordering of $P$ . Thus, the polytope can be created given a poset and a partial order.^[7]

The volume of the order polytope is given by the leading coefficient of the order polynomial, which is the number of linear extensions of the poset $P$ divided by $p!$ .^[7]^[8]

References

↑ Stanley, Richard P. (1972). Ordered structures and partitions. Providence, Rhode Island: American Mathematical Society.
↑ Stanley, Richard P. (1970). "A chromatic-like polynomial for ordered sets". Proc. Second Chapel Hill Conference on Combinatorial Mathematics and Its Appl.: 421&ndash, 427.
↑ Stanley, Richard P. (1973). "Acyclic orientations of graphs". Discrete Math. 5: 171–178.
↑ Beck, Matthias; Robins, Sinai (2015). Computing the continuous discretely. New York: Springer. pp. 64–72. ISBN 978-1-4939-2968-9.
↑ Karzanov, Alexander; Khachiyan, Leonid (1991). "On the conductance of Order Markov Chains". Order. 8: 7&ndash, 15. doi:10.1007/BF00385809.
↑ Brightwell, Graham; Winkler, Peter (1991). "Counting linear extensions". Order. 8: 225&ndash, 242. doi:10.1007/BF00383444.
1 2 Stanley, Richard P. (1986). "Two poset polytopes". Discrete Comput. Geom. doi:10.1007/BF02187680.
↑ Linial, Nathan (1984). "The information-theoretic bound is good for merging". SIAM J. Comput. 13: 795&ndash, 801. doi:10.1137/0213049.
Kahn, Jeff; Kim, Jeong Han (1995). "Entropy and sorting". Journal of Computer and System Sciences. 51: 390&ndash, 399. doi:10.1145/129712.129731.

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[1] Stanley, Richard P. (1972). Ordered structures and partitions. Providence, Rhode Island: American Mathematical Society.

[2] Stanley, Richard P. (1970). "A chromatic-like polynomial for ordered sets". Proc. Second Chapel Hill Conference on Combinatorial Mathematics and Its Appl.: 421&ndash, 427.

[3] Stanley, Richard P. (1973). "Acyclic orientations of graphs". Discrete Math. 5: 171–178.

[4] Beck, Matthias; Robins, Sinai (2015). Computing the continuous discretely. New York: Springer. pp. 64–72. ISBN 978-1-4939-2968-9.

[5] Karzanov, Alexander; Khachiyan, Leonid (1991). "On the conductance of Order Markov Chains". Order. 8: 7&ndash, 15. doi:10.1007/BF00385809.

[6] Brightwell, Graham; Winkler, Peter (1991). "Counting linear extensions". Order. 8: 225&ndash, 242. doi:10.1007/BF00383444.

[Stanley1986-7] 1 2 Stanley, Richard P. (1986). "Two poset polytopes". Discrete Comput. Geom. doi:10.1007/BF02187680.

[8] Linial, Nathan (1984). "The information-theoretic bound is good for merging". SIAM J. Comput. 13: 795&ndash, 801. doi:10.1137/0213049.
Kahn, Jeff; Kim, Jeong Han (1995). "Entropy and sorting". Journal of Computer and System Sciences. 51: 390&ndash, 399. doi:10.1145/129712.129731.