Interval order

 ( Order Theory ) 

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 set poset $P\; =\; (X,\; \backslash leq)$ is an interval order if and only if there exists a bijection from $X$ to a set of real intervals, so $x\_i\; \backslash mapsto\; (\backslash ell\_i,\; r\_i)$, such that for any $x\_i,\; x\_j\; \backslash in\; X$ we have in $P$ exactly when .

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 $(2+2)$-free posets . Fully written out, this means that for any two pairs of elements $a\; >\; b$ and $c\; >\; d$ one must have $a\; >\; d$ or $c\; >\; b$.

The subclass of interval orders obtained by restricting the intervals to those of unit length, so they all have the form $(\backslash ell\_i,\; \backslash ell\_i\; +\; 1)$, is precisely the .

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

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

Interval orders' practical applications include modelling evolution of species and archeological histories of pottery styles.This is an example.

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Interval orders and dimension An important parameter of partial orders is order dimension: the dimension of a partial order $P$ is the least number of whose intersection is $P$. 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.

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 $P\; =\; (X,\; \backslash leq)$ is the least integer $k$ for which there exist interval orders $\backslash preceq\_1,\; \backslash ldots,\; \backslash preceq\_k$ on $X$ with $x\; \backslash leq\; y$ exactly when $x\; \backslash preceq\_1\; y,\; \backslash ldots,$ and $x\; \backslash preceq\_k\; y$. The interval dimension of an order is never greater than its order dimension.

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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 $2n$ . These are the involutions with no so-called left- or right-neighbor nestings where, for any involution $f$ on $2n$, a left nesting is an $i\; \backslash in\; 2n$ such that and a right nesting is an $i\; \backslash in\; 2n$ such that .

Such involutions, according to semi-length, have ordinary generating function

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

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

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

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Notes

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• (2025). 9780521784511, Cambridge University Press. ISBN 9780521784511

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Further reading

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