Affine Balanced Labellings

• Fomin, Greene, Reiner, Shimozono - Balanced labellings and Schubert polynomials - http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.12.163&rep=rep1&type=pdf
• Edelman, Greene - Balanced tableaux - http://www.sciencedirect.com/science/article/pii/0001870887900636

• When is an affine diagram connected?
  • Answer: An affine digram is connected if and only if it is a diagonal shift of finite permutation diagram.

Classification of (Affine) Permutation Diagrams by Local Conditions

IDEAS

1. Only look at $3 \times 3$ sublattice.
   1. Among $2^9 = 512$ possible $3 \times 3$ patterns, 230 of them appears in permutations of size $\geq 6$. The actual number of patterns appearing in permutations of size 3, 4, 5, 6, 7, 8 is 6, 45, 143, 230, 230, 230.

What are the restrictions?

1. NW condition