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
- 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 $n \geq 2k$ is http://oeis.org/A048163. (verified: 2, 14, 230, 6902, 329462) http://arxiv.org/pdf/1103.4884v1.pdf (Poly-Bernoulli numbers and lonesum matrices) http://www.sciencedirect.com/science/article/pii/S0012365X1200194X (Enumeration of (0,1)-matrices avoiding some 2×2 matrices)
- Conjecture: These numbers are also equal to the number of all $k\times k$ NW-patterns. (verified: 2, 14, 230, 6902, 329462, 22934774 are the number of NW patterns.)
- Consider the matrix as an adjacency matrix of a graph, make a "diagram graph"
What are the restrictions?
- 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.
3150000 permutations examined. 329346 patterns so far. 3155000 permutations examined. 329386 patterns so far. 3160000 permutations examined. 329422 patterns so far. 3165000 permutations examined. 329448 patterns so far. 3170000 permutations examined. 329462 patterns so far. 3175000 permutations examined. 329462 patterns so far. 3180000 permutations examined. 329462 patterns so far. 3185000 permutations examined. 329462 patterns so far.