Recursive criterion for record-equivalent patterns

Establish that two patterns π,σ ∈ S_k are record-equivalent if and only if the four stated conditions hold: (1) Rec(π)=Rec(σ); (2) |Av_n(π)|=|Av_n(σ)| for all n≥k; (3) if m_π=m_σ, then for every i≠m_π, the deletion minors π^{(i)} and σ^{(i)} are record-equivalent in S_{k−1}; and (4) if k≥4, Rec(π)=Rec(σ)=[k−1], and m_π=m_σ=k−1, then π(k)=σ(k).

Background

The paper proves that the record-set statistic is equidistributed on Av_n(132) and Av_n(231), yielding identical forward-stability distributions. Motivated by this, the authors propose a general recursive criterion to decide when two patterns define avoidance classes with the same record-set distribution.

Resolving this conjecture would classify when avoidance classes are record-equivalent and, via the Hardt–Wallach max formula, when they induce identical forward stability distributions for Schubert products.

References

Conjecture [Recursive criterion for record-equivalence] Fix k≥ 2 and let π,σ∈ S_k. Then π and σ are record-equivalent if and only if all of the following hold: (1) Record-set agreement at rank k: Rec(π)=Rec(σ). (2) Wilf-equivalence at rank k: |Av_n(π)|=|Av_n(σ)| for all n≥ k. (3) Non-max deletion minors are record-equivalent: if m_π=m_σ, then for every i∈[k] with i≠ m_π, π{(i)} and σ{(i)} are record-equivalent in S_{k-1}. (4) Terminal step when the maximum sits at k-1: if k≥ 4 and Rec(π)=Rec(σ)=[k-1] and m_π=m_σ=k-1, then π(k)=σ(k).

— The record statistic and forward stability of Schubert products  (2604.02964 - Hardt et al., 3 Apr 2026) in Section 8 (Record equivalence), Conjecture [Recursive criterion for record-equivalence]