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 poset $P=(X,\leq )$ is an interval order if and only if there exists a bijection from $X$ to a set of real intervals, so $x_{i}\mapsto (\ell _{i},r_{i})$ , such that for any $x_{i},x_{j}\in X$ we have $x_{i}<x_{j}$ in $P$ exactly when $r_{i}<\ell _{j}$ . Such posets may be equivalently characterized as those with no induced subposet isomorphic to the pair of two element chains, the $(2+2)$ free posets .^[1]

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

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

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

Interval dimension

Unsolved problem in mathematics:

What is the complexity of determining the order dimension of an interval order?
(more unsolved problems in mathematics)

The interval dimension of a partial order can be defined as the minimal number of interval order extensions realizing this order, in a similar way to the definition of the order dimension which uses linear extensions. The interval dimension of an order is always less than its order dimension,^[2] but interval orders with high dimensions are known to exist. While the problem of determining the order dimension of general partial orders is known to be NP-hard, the complexity of determining the order dimension of an interval order is unknown.^[3]

Combinatorics

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

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

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

Hence the number of unlabeled interval orders of size $n$ is given by the coefficient of $t^{n}$ in the expansion of $F(t)$ .

1, 2, 5, 15, 53, 217, 1014, 5335, 31240, 201608, 1422074, 10886503, 89903100, 796713190, 7541889195, 75955177642, … (sequence A022493 in the OEIS)

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, MR 2652101, doi:10.1016/j.jcta.2009.12.007 .
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, MR 1259007, doi:10.1137/S089548019121885X .
Fishburn, Peter C. (1970), "Intransitive indifference with unequal indifference intervals", Journal of Mathematical Psychology, 7 (1): 144–149, MR 0253942, doi:10.1016/0022-2496(70)90062-3 .
Zagier, Don (2001), "Vassiliev invariants and a strange identity related to the Dedekind eta-function", Topology, 40 (5): 945–960, MR 1860536, doi:10.1016/s0040-9383(00)00005-7 .

Additional reading

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

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