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Combinatorics of generalized Dyck and Motzkin paths
Phys. Rev. E 106, 044123 – Published 14 October, 2022
DOI: https://doi.org/10.1103/PhysRevE.106.044123
Abstract
We relate the combinatorics of periodic generalized Dyck and Motzkin paths to the cluster coefficients of particles obeying generalized exclusion statistics, and obtain explicit expressions for the counting of paths with a fixed number of steps of each kind at each vertical coordinate. A class of generalized compositions of the integer path length emerges in the analysis.
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References (17)
- D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14, 2239 (1976).
- Ch. Kreft, Explicit computation of the discriminant for the Harper equation with rational flux, SFB 288 Preprint No. 89, TU-Berlin, 1993 (unpublished).
- S. Ouvry and S. Wu, The algebraic area of closed lattice random walks, J. Phys. A: Math. Theor. 52, 255201 (2019).
- S. Ouvry and A. P. Polychronakos, Exclusion statistics and lattice random walks, Nucl. Phys. B 948, 114731 (2019); Lattice walk area combinatorics, some remarkable trigonometric sums and Apéry-like numbers, 960, 115174 (2020).
- B. Hopkins and S. Ouvry, Combinatorics of multicompositions, in Proceedings of the Combinatorial and Additive Number Theory Conference (Springer, Berlin, 2020).
- C. Krattenthaler, Permutations with restricted patterns and Dyck paths, Adv. Appl. Math. 27, 510 (2001).
- G. M. Cicuta, M. Contedini and L. Molinari, Enumeration of simple random walks and tridiagonal matrices, J. Phys. A: Math. Gen. 35, 1125 (2002).
- L. Gan, S. Ouvry, and A. P. Polychronakos, Algebraic area enumeration of random walks on the honeycomb lattice, Phys. Rev. E 105, 014112 (2022).
- R. Oste and J. Van der Jeugt, Motzkin paths, Motzkin polynomials and recurrence relations, Electron. J. Combin. 22, P2.8 (2015).
- S. Ouvry and A. P. Polychronakos, Hamiltonian and exclusion statistics approach to discrete forward-moving paths, Phys. Rev. E 104, 014143 (2021).
- A. P. Polychronakos, Length and area generating functions for height-restricted Motzkin meanders, Phys. Rev. E 105, 024102 (2022).
- S. Ouvry and A. P. Polychronakos, Algebraic area enumeration for open latticewalks, arXiv:2206.12428.
- R. Brak, G. K. Iliev, and T. Prellberg, An infinite family of adsorption models and restricted Lukasiewicz paths, J. Stat. Phys. 145, 669 (2011).
- S. Bravyi, L. Caha, R. Movassagh, D. Nagaj, and P. W. Shor, Criticality without Frustration for Quantum Spin-1 Chains, Phys. Rev. Lett. 109, 207202 (2012).
- R. J. Baxter, A simple solvable Hamiltonian, Phys. Lett. A 140, 155 (1989); Superintegrable chiral Potts model: Thermodynamic properties, an inverse model and a simple associated Hamiltonian, J. Stat. Phys. 57, 1 (1989); Transfer matrix functional relations for the generalized model, 117, 1 (2004).
- P. Fendley, Free parafermions, J. Phys. A: Math. Theor. 47, 075001 (2014); F. C. Alcaraz and R. A. Pimenta, Free fermionic and parafermionic quantum spin chains with multispin interactions, Phys. Rev. B 102, 121101(R) (2020).
- E. Posner, K. Hatch, and M. Ly, Presentation of the Motzkin monoid, arXiv:1301.4518.
