Difference between revisions of "Affine Balanced Labellings"
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# Only look at $3 \times 3$ sublattice. | # 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. | ## 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 $k\times k$ | + | ## Conjecture: The number of $k\times k$ patterns in permutation diagrams of size $n$ is http://oeis.org/A048163. (verified: 2, 14, 230) |
=== What are the restrictions? === | === What are the restrictions? === | ||
# NW condition | # NW condition | ||
Revision as of 22:37, 16 January 2013
- 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 $n$ is http://oeis.org/A048163. (verified: 2, 14, 230)
What are the restrictions?
- NW condition