Abstract
This extended abstract proves that the number of fully packed loop configurations whose link pattern consists of two noncrossing matchings separated by m nested arches is a polynomial in m. This was conjectured by Zuber (2004) and for large values of m proved by Caselli et al. (2004)
Highlights
Razumov and Stroganov conjectured in [5] a relation between fully packed loop configurations and the ground state vector in the O(1) loop model
Theorem 1.1 ([11, Conjecture 7]) For noncrossing matchings π ∈ N Cn and π ∈ N Cn and an integer m the number of FPLs with link pattern (π)mπ is a polynomial in m of degree |λ(π)| + |λ(π )| with leading coefficient dim(λ(π)) dim(λ(π |λ(π)|!|λ(π )|!
We can assign to every FPL F a noncrossing matching π(F ) by setting π(F )(i) the label of the external edge which is connected to the i-th external edge for 1 ≤ i ≤ 2n, and call π(F ) the link pattern of F
Summary
Razumov and Stroganov conjectured in [5] a relation between fully packed loop configurations (short FPLs) and the ground state vector in the O(1) loop model This conjectural connection made it possible to come up with many new conjectures concerning FPLs with certain link patterns, see e. With the proof of the RazumovStroganov Conjecture in [1], it was possible to use new methods – namely the theory of wheel polynomials which was developed in [2], [9] – for solving open problems concerning FPLs. Using integral wheel polynomials, Fonseca and Zinn-Justin reproved in [4] Theorem 1.1 for the special case of one matching. 1365–8050 c 2016 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France
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