Related papers: Counting linear extensions of restricted posets
Partially ordered sets (posets) are fundamental combinatorial objects with important applications in computer science. Perhaps the most natural algorithmic task, given a size-$n$ poset, is to compute its number of linear extensions. In 1991…
Let $(\mathcal{P},\leqslant)$ be a finite poset. Define the numbers $a_1,a_2,\ldots$ (respectively, $c_1,c_2,\ldots$) so that $a_1+\ldots+a_k$ (respectively, $c_1+\ldots+c_k$) is the maximal number of elements of $\mathcal{P}$ which may be…
We give a broad survey of inequalities for the number of linear extensions of finite posets. We review many examples, discuss open problems, and present recent results on the subject. We emphasize the bounds, the equality conditions of the…
Counting linear extensions is a fundamental problem in poset theory. It is known to be #P-complete, with polynomial-time formulas available in special cases. In this work, we develop new recursive formulas for counting linear extensions of…
Kahn and Kim (J. Comput. Sci., 1995) have shown that for a finite poset $P$, the entropy of the incomparability graph of $P$ (normalized by multiplying by the order of $P$) and the base-$2$ logarithm of the number of linear extensions of…
We introduce several new constructions of finite posets with the number of linear extensions given by generalized continued fractions. We apply our results to the problem of the minimum number of elements needed for a poset with a given…
Joret et al. proved that posets with cover graphs of treewidth at most 2 have dimension at most 1276. Their proof is long and very complex. We give a short and much simpler proof that the dimension of such posets is at most 12.
The cross--product conjecture (CPC) of Brightwell, Felsner and Trotter (1995) is a two-parameter quadratic inequality for the number of linear extensions of a poset $P= (X, \prec)$ with given value differences on three distinct elements in…
The linear extension diameter of a finite poset P is the maximum distance between a pair of linear extensions of P, where the distance between two linear extensions is the number of pairs of elements of P appearing in different orders in…
Given a finite poset P, we consider pairs of linear extensions of P with maximal distance, where the distance between two linear extensions L_1, L_2 is the number of pairs of elements of P appearing in different orders in L_1 and L_2. A…
The queue number of a poset is the queue number of its cover graph when the vertex order is a linear extension of the poset. Heath and Pemmaraju conjectured that every poset of width $w$ has queue number at most $w$. The conjecture has been…
A grid poset -- or grid for short -- is a product of chains. We ask, what does a random linear extension of a grid look like? In particular, we show that the average "jump number," i.e., the number of times that two consecutive elements in…
A linear extension of a poset $P$ is a permutation of the elements of the set that respects the partial order. Let $L(P)$ denote the number of linear extensions. It is a #P complete problem to determine $L(P)$ exactly for an arbitrary…
In this paper we consider the classical problem of computing linear extensions of a given poset which is well known to be a difficult problem. However, in our setting the elements of the poset are multivariate polynomials, and only a small…
Motivated by the study of the dimension of random posets, it was conjectured by Bollob\'as and Brightwell in 1997 that if $P$ is a finite poset whose cover graph contains at most one cycle then its order dimension is at most $3$. In this…
In 1977, Trotter and Moore proved that a poset has dimension at most $3$ whenever its cover graph is a forest, or equivalently, has treewidth at most $1$. On the other hand, a well-known construction of Kelly shows that there are posets of…
We introduce a class of posets, which includes both ribbon posets (skew shapes) and $d$-complete posets, such that their number of linear extensions is given by a determinant of a matrix whose entries are products of hook lengths. We also…
The minimum number of elements needed for a poset to have exactly n linear extensions is at most 2sqrt{n}. In a special case, the bound can be improved to sqrt{n}.
Sign imbalance is a statistic on posets which counts the difference between the number of even and odd linear extensions. We prove complexity results about the sign imbalance and parity of linear extensions, focusing on the representative…
We investigate the queue number of posets in terms of their width, that is, the maximum number of pairwise incomparable elements. A long-standing conjecture of Heath and Pemmaraju asserts that every poset of width w has queue number at most…