A Note on Antichains of Words

# a 29 Integers 9 ( 2009 ) , 353 - 374 Combinatorial Properties of the Antichains Of

In this paper we study some combinatorial properties of the antichains of a garland, or double fence. Specifically, we encode the order ideals and the antichains in terms of words of a regular language, we obtain several enumerative properties (such as generating series, recurrences and explicit formulae), we consider some statistics leading to Riordan matrices, we study the relations between t...

On Antichains in Product Posets

A corollary of Hilbert’s basis theorem is that any antichain in the set of n-dimensional vectors with non-negative entries is finite. In other words, any antichain in the poset given by cartesian powers of semi-infinite chains is finite. We study several variations of this result. We provide necessary and sufficient conditions for antichains in the cartesian product of posets to be finite or bo...

A Note on Operators of Deletion and Contraction for Antichains

Counting Antichains and Linear Extensions in Generalizations of the Boolean Lattice

In 1897, Dedekind [6] posed the problem of estimating the number of antichains in the Boolean lattice; in particular, he asked whether the logarithm of the number is asymptotic to the size of the middle layer of the n-dimensional Boolean lattice Bn. Although Kleitman confirmed the truth of this conjecture in 1969 [13], enumerating antichains in Bn has continued to generate interest in the mathe...

Generalized Factors of Words

We introduce and study relations on words which generalize the factor relation, being restrictions of the subword relation. We give an equivalent condition for the nite basis property for these relations which generalizes the well-known theorem of Higman. Some language-theoretic gaps for in nite antichains are also presented.