Blocking Ideals: a method for filtering linear extensions of a finite poset

TL;DR

This paper introduces a novel method using blocking ideals in the lattice of order ideals to efficiently count linear extensions of finite posets, aiding in probability calculations related to the 1/3-2/3-conjecture.

Contribution

It presents an alternative approach to counting linear extensions via blocking ideals, applicable to all finite posets, and demonstrates its use in calculating probabilities in cell posets.

Findings

Derived explicit formulas for probabilities in two-row partition posets.

Connected the method to classical formulas like hook-length and Jacobi-Trudi.

Calculated limit probabilities for fixed cell pairs as arm-lengths grow.

Abstract

The standard notion of poset probability of a finite poset P involves calculating, for incomparable $\alpha$, $\beta$ in P, the number of linear extensions of P for which $\alpha$ precedes $\beta$. The fraction of those linear extensions among all linear extensions of P is the probability that $\alpha < \beta$. The question of whether there is always a pair $\alpha, \beta$ such that this probability lies between 1/3 and 2/3, in any poset P (that is not a chain) is the famous "1/3-2/3-conjecture". A general way of counting linear extensions of P for which $\alpha$ precedes $\beta$ is to count linear extensions of the poset obtained by adding the relation $(\alpha,\beta)$, and its transitive consequences. For chain-products, and more generally for partition posets, lattice-path methods can be used to count the number of those linear extensions. We present an alternative approach to find the pertinent linear extensions. It relies on finding the "blocking ideals" in $J(P)$, where $J(P)$ is the lattice of order ideals in P. This method works for all finite posets. We illustrate this method by using blocking ideals to find explicit formulas of poset probabilities in cell posets $P_{\lambda}$ of two-row partitions. Well-known formulae such as the hook-length formula for $f^\lambda$, the number of standard Young tableaux on a partition $\lambda$, and the corresponding determinental formula by Jacobi-Trudi-Aitken for $f^{\lambda / \mu}$, the number of standard Young tableaux on a skew partition $\lambda / \mu$, are used along the way. We also calculate the limit probabilities when the elements $\alpha,\beta$ are fixed cells, but the arm-lengths tend to infinity.