Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T09:02:59.572Z Has data issue: false hasContentIssue false
  • English
  • Français

Multivariate Eulerian Polynomials and Exclusion Processes

Part of: Combinatorics Combinatorial probability

Published online by Cambridge University Press:  18 March 2016

P. BRÄNDÉN ,
M. LEANDER  and
M. VISONTAI
P. BRÄNDÉN
Affiliation:
Department of Mathematics, Royal Institute of Technology, SE-100 44, Stockholm, Sweden (e-mail: pbranden@kth.se, visontai@kth.se)
M. LEANDER
Affiliation:
Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden (e-mail: madde@math.su.se)
M. VISONTAI
Affiliation:
Department of Mathematics, Royal Institute of Technology, SE-100 44, Stockholm, Sweden (e-mail: pbranden@kth.se, visontai@kth.se)
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a new combinatorial interpretation of the stationary distribution of the (partially) asymmetric exclusion process on a finite number of sites in terms of decorated alternative trees and coloured permutations. The corresponding expressions of the multivariate partition functions are then related to multivariate generalisations of Eulerian polynomials for coloured permutations considered recently by N. Williams and the third author, and others. We also discuss stability and negative dependence properties satisfied by the partition functions.

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05A19: Combinatorial identities, bijective combinatorics
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 4 , July 2016 , pp. 486 - 499
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Borcea, J. and Brändén, P. (2009) The Lee–Yang and Pólya–Schur programs I: Linear operators preserving stability. Invent. Math. 177 541–569.Google Scholar
[2] Borcea, J., Brändén, P. and Liggett, T. M. (2009) Negative dependence and the geometry of polynomials. J. Amer. Math. Soc. 22 521–567.Google Scholar
[3] Brenti, F. (1994) q-Eulerian polynomials arising from Coxeter groups. European J. Combin. 15 417–441.Google Scholar
[4] Chen, W. Y. C., Hao, R. X. J. and Yang, H. R. L. Context-free grammars and multivariate stable polynomials over Stirling permutations. arXiv:1208.1420 Google Scholar
[5] Corteel, S. (2007) Crossings and alignments of permutations. Adv. Appl. Math. 38 149–163.Google Scholar
[6] Corteel, S. and Williams, L. K. (2007) Tableaux combinatorics for the asymmetric exclusion process. Adv. Appl. Math. 39 293–310.Google Scholar
[7] Corteel, S. and Williams, L. K. (2011) Tableaux combinatorics for the asymmetric exclusion process and Askey–Wilson polynomials. Duke Math. J. 159 385–415.Google Scholar
[8] Derrida, B., Evans, M. R., Hakim, V. and Pasquier, V. (1993) Exact solution of a 1D asymmetric exclusion model using a matrix formulation. J. Phys. A 26 1493–1517.Google Scholar
[9] Brändén, P., Haglund, J., Visontai, M. and Wagner, D. G. (2011) Proof of the monotone column permanent conjecture. In Notions of Positivity and the Geometry of Polynomials, Trends in Mathematics, Birkhäuser, pp. 63–78.Google Scholar
[10] Choe, Y.-B., Oxley, J. G., Sokal, A. D. and Wagner, D. G. (2004) Homogeneous multivariate polynomials with the half-plane property. Adv. Appl. Math. 32 (special issue on the Tutte polynomial) 88–187.Google Scholar
[11] Ehrenborg, R. and Steingrímsson, E. (2000) The excedance set of a permutation. Adv. Appl. Math. 24 284–299.CrossRefGoogle Scholar
[12] Frobenius, G. (1910) Über die Bernoulli'schen Zahlen und die Euler'schen Polynome, Vol. 2 of Sitzungs-berichte der Königlich Preussischen Akademie der Wissenschaften.Google Scholar
[13] Haglund, J. and Visontai, M. (2012) Stable multivariate Eulerian polynomials and generalized Stirling permutations. European J. Combin. 33 477–487.Google Scholar
[14] Nadeau, P. (2011) The structure of alternative tableaux. J. Combin. Theory Ser. A 118 1638–1660.Google Scholar
[15] Hitczenko, P. and Janson, S. (2014) Weighted random staircase tableaux. Combin. Probab. Comput. 23 1114–1147.CrossRefGoogle Scholar
[16] Josuat-Vergès, M. (2011) Combinatorics of the three-parameter PASEP partition function. Electron. J. Combin. 18 P22.CrossRefGoogle Scholar
[17] Liggett, T. M. (1975) Ergodic theorems for the asymmetric simple exclusion process. Trans. Amer. Math. Soc. 213 237–261.CrossRefGoogle Scholar
[18] Pemantle, R. (2000) Towards a theory of negative dependence. J. Math. Phys. 41 1371–1390.Google Scholar
[19] Pemantle, R. and Peres, Y. (2014) Concentration of Lipschitz functionals of determinantal and other strong Rayleigh measures. Combin. Probab. Comput. 23 140–160.CrossRefGoogle Scholar
[20] Savage, C. D. and Visontai, M. (2015) The s-Eulerian polynomials have only real zeros. Trans. Amer. Math. Soc. 367 1441–1466.Google Scholar
[21] Steingrímsson, E. (1994) Permutation statistics of indexed permutations. European J. Combin. 15 187–205.Google Scholar
[22] Viennot, X. (2007) Alternative tableaux, permutations and partially asymmetric exclusion process. In Proc. Workshop ‘Statistical Mechanics and Quantum-Field Theory Methods in Combinatorial Enumeration’, Isaac Newton Institute, April 2007. www.newton.ac.uk/webseminars/pg+ws/2008/csm/csmw04/0423/viennot/ Google Scholar
[23] Visontai, M. and Williams, N. (2013) Stable multivariate W-Eulerian polynomials. J. Combin. Theory Ser. A 120 1929–1945.Google Scholar
[24] Wagner, D. G. (2011) Multivariate stable polynomials: theory and applications. Bull. Amer. Math. Soc. 48 53–84.Google Scholar