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.
   • 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.
   • Conjecture: The number of $k\times k$ patterns in permutation diagrams of size $n$ is http://oeis.org/A048163. (verified: 2, 14, 230, 6902)

What are the restrictions?

1. NW condition

671000 permutations examined. 267910 patterns so far. 672000 permutations examined. 267940 patterns so far. 673000 permutations examined. 267955 patterns so far. 674000 permutations examined. 267956 patterns so far. 675000 permutations examined. 267956 patterns so far. 676000 permutations examined. 267956 patterns so far. 677000 permutations examined. 267956 patterns so far. 678000 permutations examined. 267956 patterns so far.