The lattice of antichains of intervals: Dyck paths, level sets, and sweep dimension

Abstract

The Clarke-Cormack-Burkowski lattice $\mathcal{E}_n$ is the lattice of antichains of intervals of the chain $\{0,1,\dots,n-1\}$, the algebraic structure underlying minimal-interval semantics, a query semantics used in information retrieval. We give an explicit, elementary anti-isomorphism between $\mathcal{E}_n$ minus its top element and the Stanley lattice of Dyck words of semilength $n+1$ (an anti-isomorphism whose existence is implicit in work of Ferrari and Munarini), under which the rank of an antichain is $\binom{n+1}{2}$ minus the area of the associated Dyck word. Hence the rank generating function of $\mathcal{E}_n\setminus\{\top\}$ is the reverse of the Carlitz-Riordan $q$-Catalan polynomial. Using a local limit theorem for the area of random-walk excursions, we confirm unconditionally an asymptotic formula of Takács: the size $λ_n$ of the largest level set of $\mathcal{E}_n$ satisfies $λ_n\simκ\,4^{n+1}/(n+1)^3$, where $κ\approx1.0807$. Finally, for the antichains of intervals of an arbitrary finite poset $P$ we study the sweep dimension, the minimum number of linear orders needed to separate every interval from every element outside it. It equals the Dushnik-Miller dimension of the lattice of convex subsets of $P$, it is the least $k$ for which $P$ maps into $\mathbb{R}^k$ so that every interval is the preimage of an upper orthant, and it lies between $\max(2,w)$ and $2w$, where $w$ is the width of $P$, both bounds being tight. Antichains of intervals embed, preserving joins, into $k$-dimensional staircase regions.