[Paper Review] Sorting probability for large Young diagrams

Swee Hong Chan, Igor Pak+1 more|arXiv (Cornell University)|May 17, 2020

Advanced Combinatorial Mathematics37 references4 citations

TL;DR

This paper establishes sharp asymptotic upper bounds on the sorting probability δ(P) for large Young diagrams and skew Young diagrams with a bounded number of rows. Using the hook-length formula and Schur function techniques, it proves δ(Pλ) ≤ Cα/√n for Thoma-shaped partitions and δ(Pλ) ≤ Cd,ε/√n for ε-thick partitions, providing the first such results in this regime and advancing the 1/3–2/3 conjecture for structured posets.

ABSTRACT

For a finite poset $P=(X,\prec)$, let $\mathcal{L}_P$ denote the set of linear extensions of $P$. The sorting probability $δ(P)$ is defined as \[δ(P) \, := \, \min_{x,y\in X} \, \bigl| \mathbf{P} \, [L(x)\leq L(y) ] \ - \ \mathbf{P} \, [L(y)\leq L(x) ] \bigr|\,, \] where $L \in \mathcal{L}_P$ is a uniform linear extension of $P$. We give asymptotic upper bounds on sorting probabilities for posets associated with large Young diagrams and large skew Young diagrams, with bounded number of rows.

Motivation & Objective

• To establish asymptotic upper bounds on the sorting probability δ(P) for large Young diagrams and skew Young diagrams with bounded row counts.
• To investigate the 1/3–2/3 conjecture in the context of structured posets arising from Young diagrams.
• To extend known results on linear extensions and sorting probabilities to the asymptotic regime of large diagrams.
• To analyze the behavior of δ(P) using combinatorial tools such as the hook-length formula and Schur functions.

Proposed method

• The authors use the hook-length formula and Schur functions to estimate the number of linear extensions of Young diagram posets.
• They apply the Naruse hook-length formula (NHLF) to compute the number of standard Young tableaux for skew shapes.
• A quantitative pigeonhole principle argument is used to bound the difference in probabilities of orderings of pairs of elements.
• Interval decompositions are introduced to refine the pigeonhole argument and control variance in linear extension probabilities.
• The analysis is conducted under assumptions of ε-thinness or ε-smoothness to ensure uniformity in row sizes.
• The results are extended to cases where deviations from ideal shapes are bounded by a constant K, maintaining the O(1/√n) decay rate.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the sorting probability δ(P) for large Young diagrams with a fixed number of rows?
• RQ2 Can the 1/3–2/3 conjecture be strengthened to an O(1/√n) upper bound for structured posets like Young diagrams?
• RQ3 How do the properties of the partition shape—such as row size uniformity or thickness—affect the sorting probability?
• RQ4 To what extent can the hook-length formula and Schur function methods be used to bound sorting probabilities in asymptotic regimes?
• RQ5 Is the ε-thickness assumption necessary for the O(1/√n) bound, or can it be removed?

Key findings

• For any Thoma sequence α ∈ ℝ₊ᵈ with |α| = 1, the sorting probability satisfies δ(Pλ) ≤ Cα/√n for λ ≃ αn, where Cα is a universal constant depending on α.
• For ε-thick partitions λ ⊢ n with d rows and λd ≥ εn, δ(Pλ) ≤ Cd,ε/√n, with Cd,ε universal for fixed d and ε.
• The bound δ(Pλ) = O(1/√n) holds uniformly across all such partitions, even when row sizes vary moderately.
• The results generalize previous bounds and represent the first asymptotic estimates for sorting probabilities in this class of posets.
• The ε-thickness assumption is likely not necessary, as the O(1/n) bound holds in the extreme case where λ₂ = O(1), suggesting the O(1/√n) bound may extend more broadly.
• The upper bound can be extended to cases with bounded deviation from ideal shapes, with the constant depending on the deviation bound K.