Interval order

In mathematics, especially order theory, the interval order for a collection of intervals on the real line is the partial order corresponding to their left-to-right precedence relation—one interval, I₁, being considered less than another, I₂, if I₁ is completely to the left of I₂. More formally, a countable poset ${\displaystyle P=(X,\leq )}$ is an interval order if and only if there exists a bijection from ${\displaystyle X}$ to a set of real intervals, so ${\displaystyle x_{i}\mapsto (\ell _{i},r_{i})}$, such that for any ${\displaystyle x_{i},x_{j}\in X}$ we have ${\displaystyle x_{i}<x_{j}}$ in ${\displaystyle P}$ exactly when ${\displaystyle r_{i}<\ell _{j}}$.

Such posets may be equivalently characterized as those with no induced subposet isomorphic to the pair of two-element chains, in other words as the ${\displaystyle (2+2)}$-free posets .^[1] Fully written out, this means that for any two pairs of elements ${\displaystyle a>b}$ and ${\displaystyle c>d}$ one must have ${\displaystyle a>d}$ or ${\displaystyle c>b}$.

The subclass of interval orders obtained by restricting the intervals to those of unit length, so they all have the form ${\displaystyle (\ell _{i},\ell _{i}+1)}$, is precisely the semiorders.

The complement of the comparability graph of an interval order (${\displaystyle X}$, ≤) is the interval graph ${\displaystyle (X,\cap )}$.

Interval orders should not be confused with the interval-containment orders, which are the inclusion orders on intervals on the real line (equivalently, the orders of dimension ≤ 2).

Interval orders and dimension

An important parameter of partial orders is order dimension: the dimension of a partial order ${\displaystyle P}$ is the least number of linear orders whose intersection is ${\displaystyle P}$. For interval orders, dimension can be arbitrarily large. And while the problem of determining the dimension of general partial orders is known to be NP-hard, determining the dimension of an interval order remains a problem of unknown computational complexity.^[2]

A related parameter is interval dimension, which is defined analogously, but in terms of interval orders instead of linear orders. Thus, the interval dimension of a partially ordered set ${\displaystyle P=(X,\leq )}$ is the least integer ${\displaystyle k}$ for which there exist interval orders ${\displaystyle \preceq _{1},\ldots ,\preceq _{k}}$ on ${\displaystyle X}$ with ${\displaystyle x\leq y}$ exactly when ${\displaystyle x\preceq _{1}y,\ldots ,}$ and ${\displaystyle x\preceq _{k}y}$. The interval dimension of an order is never greater than its order dimension.^[3]

Combinatorics

In addition to being isomorphic to ${\displaystyle (2+2)}$-free posets, unlabeled interval orders on ${\displaystyle [n]}$ are also in bijection with a subset of fixed-point-free involutions on ordered sets with cardinality ${\displaystyle 2n}$ .^[4] These are the involutions with no so-called left- or right-neighbor nestings where, for any involution ${\displaystyle f}$ on ${\displaystyle [2n]}$, a left nesting is an ${\displaystyle i\in [2n]}$ such that ${\displaystyle i<i+1<f(i+1)<f(i)}$ and a right nesting is an ${\displaystyle i\in [2n]}$ such that ${\displaystyle f(i)<f(i+1)<i<i+1}$.

Such involutions, according to semi-length, have ordinary generating function^[5]

${\displaystyle F(t)=\sum _{n\geq 0}\prod _{i=1}^{n}(1-(1-t)^{i}).}$

The coefficient of ${\displaystyle t^{n}}$ in the expansion of ${\displaystyle F(t)}$ gives the number of unlabeled interval orders of size ${\displaystyle n}$. The sequence of these numbers (sequence A022493 in the OEIS) begins

1, 2, 5, 15, 53, 217, 1014, 5335, 31240, 201608, 1422074, 10886503, 89903100, 796713190, 7541889195, 75955177642, …

Notes

References

• Bousquet-Mélou, Mireille; Claesson, Anders; Dukes, Mark; Kitaev, Sergey (2010), "(2+2) free posets, ascent sequences and pattern avoiding permutations", Journal of Combinatorial Theory, Series A, 117 (7): 884–909, arXiv:0806.0666, doi:10.1016/j.jcta.2009.12.007, MR 2652101, S2CID 8677150.
• Felsner, S. (1992), Interval Orders: Combinatorial Structure and Algorithms (PDF), Ph.D. dissertation, Technical University of Berlin.
• Felsner, S.; Habib, M.; Möhring, R. H. (1994), "On the interplay between interval dimension and dimension" (PDF), SIAM Journal on Discrete Mathematics, 7 (1): 32–40, doi:10.1137/S089548019121885X, MR 1259007.
• Fishburn, Peter C. (1970), "Intransitive indifference with unequal indifference intervals", Journal of Mathematical Psychology, 7 (1): 144–149, doi:10.1016/0022-2496(70)90062-3, MR 0253942.
• Zagier, Don (2001), "Vassiliev invariants and a strange identity related to the Dedekind eta-function", Topology, 40 (5): 945–960, doi:10.1016/s0040-9383(00)00005-7, MR 1860536.

Further reading

• Fishburn, Peter (1985), Interval Orders and Interval Graphs: A Study of Partially Ordered Sets, John Wiley