Mu–Welker Dyck valleys and bargraph statement

Abstract

State Mu–Welker’s Conjecture 3.9: UUDD-avoiding Dyck paths with i valleys are counted by bargraphs of semiperimeter n and width i.

Definition 1.1 (The source-side predicates).

Formalization. D5/S3/Combinatorics/DyckValleys/ValleyBargraphDefs.AvoidsUUDD, valleys, ascent, semiperimeter, and IsBargraph.

Citation. Lili Mu and Volkmar Welker (2026). Simplicial Complexes of Antichains in Root Posets and Related Combinatorics of Dyck Paths, arXiv:2609.14054v1, §3, Conjecture 3.9. URL: https://arxiv.org/abs/2609.14054.

Commentary. AvoidsUUDD is the maximal-face condition from the source’s Lemma 3.4. valleys counts adjacent DU factors. A positive height list is a bargraph; semiperimeter is length + h₁ + Σ max(hⱼ₊₁ − hⱼ, 0).

Definition 1.2 (The conjecture statement).

Formalization. D5/S3/Combinatorics/DyckValleys/ValleyBargraphDefs.claim.

The statement quantifies n ≥ 2 and 1 ≤ i ≤ n−1, equating the finite cardinality of the source-side Dyck words with the finite cardinality of positive height lists of length i and semiperimeter n. This file records the exact open target; it does not claim that the target is already resolved.

References

• Truth anchor: D5/S3/Combinatorics/DyckValleys/ValleyBargraphDefs.claim