Enumeration of 2341-avoiding distinct flattened partitions

Determine and prove that the number p_n(2341) of 2341-avoiding distinct flattened partitions of [n] equals the (n−1)-st term of OEIS sequence A307786 for every positive integer n, namely, prove that p_n(2341)=A307786(n−1) for all n≥1.

Background

The pattern 2341 appears as a singleton Wilf-equivalence class in the paper’s numerical classification of length-four patterns. Its avoidance sequence appears to agree with the shifted OEIS sequence A307786, and this agreement was confirmed numerically through n=15. However, the paper does not establish the claimed formula for all n and presents it as a conjecture.

References

If $n \geq 1$, then $p_n(2341)=A307786(n-1)$.

— Distinct flattened partitions avoiding a pattern of length four  (2610.11271 - Mansour et al., 8 Oct 2026) in Section 6, Further results, second Conjecture following Table \ref{tab:we-classes}