Interval order

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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[edit]

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-complete, the complexity of determining the order dimension of an interval order is unknown.[3]

Combinatorics[edit]

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 coefficent 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 OEIS)

Notes[edit]

References[edit]

  • Bousquet-Mélou, Mireille; Claesson, Anders; Dukes, Mark; Kitaev, Sergey (2010), "(2+2) free Posets, Ascent Sequences and Pattern Avoiding Permutations", J. Comb. Theory Ser. A (Academic Press, Inc.) 117 (7): 884–909, doi:10.1016/j.jcta.2009.12.007, ISSN 0097-3165 
  • Zagier, Don (2001), "Vassiliev invariants and a strange identity related to the Dedekind eta-function", Topology 40 (5): 945–960, ISSN 0040-9383 
  • Fishburn, Peter (1985), Interval Orders and Interval Graphs: A Study of Partially Ordered Sets, John Wiley 
  • Fishburn, Peter C. (1970), "Intransitive indifference with unequal indifference intervals", Journal of Mathematical Psychology 7 (1): 144–149 .