TY - JOUR

T1 - Asymptotics for a Class of Meandric Systems, via the Hasse Diagram of NC(n)

AU - Goulden, I. P.

AU - Nica, Alexandru

AU - Puder, Doron

N1 - Publisher Copyright:
© 2018 The Author(s) 2018. Published by Oxford University Press. All rights reserved.

PY - 2020/2/20

Y1 - 2020/2/20

N2 - We consider the framework of closed meandric systems and its equivalent description in terms of the Hasse diagrams of the lattices of non-crossing partitions NC(n). In this equivalent description, considerations on the number of components of a random meandric system of order n translate into considerations about the distance between two random partitions in NC(n). We put into evidence a class of couples $(\pi, \rho) \in \textrm{NC}(n)^{2}$ - namely the ones where $\pi $ is conditioned to be an interval partition - for which it turns out to be tractable to study distances in the Hasse diagram. As a consequence, we observe a nontrivial class of meanders (i.e., connected meandric systems), which we call meanders with shallow top and which can be explicitly enumerated. Moreover, denoting by $c-{n}$ the expected number of components for the corresponding notion of meandric system with shallow top of order n, we find the precise asymptotic $c-{n}\approx \frac{n}{3}+\frac{28}{27}$ for $n\to \infty $. Our calculations concerning expected number of components are related to the idea of taking the derivative at t = 1 in a semigroup for the operation $\boxplus $ of free probability (but the underlying considerations are presented in a self-contained way and can be followed without assuming a free probability background). Let $c-{n}^{\prime }$ denote the expected number of components of a general, unconditioned, meandric system of order n. A variation of the methods used in the shallow-top case allows us to prove that $\mathrm{lim\ inf}-{n\to \infty }c-{n}^{\prime }/n\geq 0.17$. We also note that, by a direct elementary argument, one has $\mathrm{lim\ sup}-{n\to \infty }c-{n}^{\prime }/n\leq 0.5$. These bounds support the conjecture that $c-{n}^{\prime }$ follows a regime of constant times n (where numerical experiments suggest that the constant should be ≈ 0.23).

AB - We consider the framework of closed meandric systems and its equivalent description in terms of the Hasse diagrams of the lattices of non-crossing partitions NC(n). In this equivalent description, considerations on the number of components of a random meandric system of order n translate into considerations about the distance between two random partitions in NC(n). We put into evidence a class of couples $(\pi, \rho) \in \textrm{NC}(n)^{2}$ - namely the ones where $\pi $ is conditioned to be an interval partition - for which it turns out to be tractable to study distances in the Hasse diagram. As a consequence, we observe a nontrivial class of meanders (i.e., connected meandric systems), which we call meanders with shallow top and which can be explicitly enumerated. Moreover, denoting by $c-{n}$ the expected number of components for the corresponding notion of meandric system with shallow top of order n, we find the precise asymptotic $c-{n}\approx \frac{n}{3}+\frac{28}{27}$ for $n\to \infty $. Our calculations concerning expected number of components are related to the idea of taking the derivative at t = 1 in a semigroup for the operation $\boxplus $ of free probability (but the underlying considerations are presented in a self-contained way and can be followed without assuming a free probability background). Let $c-{n}^{\prime }$ denote the expected number of components of a general, unconditioned, meandric system of order n. A variation of the methods used in the shallow-top case allows us to prove that $\mathrm{lim\ inf}-{n\to \infty }c-{n}^{\prime }/n\geq 0.17$. We also note that, by a direct elementary argument, one has $\mathrm{lim\ sup}-{n\to \infty }c-{n}^{\prime }/n\leq 0.5$. These bounds support the conjecture that $c-{n}^{\prime }$ follows a regime of constant times n (where numerical experiments suggest that the constant should be ≈ 0.23).

UR - http://www.scopus.com/inward/record.url?scp=85081915375&partnerID=8YFLogxK

U2 - 10.1093/imrn/rny044

DO - 10.1093/imrn/rny044

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AN - SCOPUS:85081915375

SN - 1073-7928

VL - 2020

SP - 983

EP - 1034

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 4

ER -