• XAVIER VIENNOT
  • Foreword
    • Preface
    • Introduction
    • Acknowledgements
    • Lectures for wide audience
  • PART I
    • Preface
    • Abstract
    • Contents
    • Ch0 Introduction to the course
    • Ch1 Ordinary generating functions
    • Ch2 The Catalan garden
    • Ch3 Exponential structures and genarating functions
    • Ch4 The n! garden
    • Ch5 Tilings, determinants and non-intersecting paths
    • Lectures related to the course
    • List of bijections
    • Index
  • PART II
    • Preface
    • Abstract
    • Contents
    • Ch1 Commutations and heaps of pieces: basic definitions
    • Ch2 Generating functions of heaps of pieces
    • Ch3 Heaps and paths, flows and rearrangements monoids
    • Ch4 Linear algebra revisited with heaps of pieces
    • Ch5 Heaps and algebraic graph theory
    • Ch6 Heaps and Coxeter groups
    • Ch7 Heaps in statistical mechanics
    • Lectures related to the course
  • PART III
    • Preface
    • Abstract
    • Contents
    • Ch0 overview of the course
    • Ch1 RSK The Robinson-Schensted-Knuth correspondence
    • Ch2 Quadratic algebra, Q-tableaux and planar automata
    • Ch3 Tableaux for the PASEP quadratic algebra
    • Ch4 Trees and tableaux
    • Ch5 Tableaux and orthogonal polynomials
    • Ch6 Extensions: tableaux for the 2-PASEP quadratic algebra
    • Lectures related to the course
    • References, comments and historical notes
  • PART IV
    • Preface
    • Introduction
    • Contents
    • Ch0 Overview of the course
    • Ch1 Paths and moments
    • Ch2 Moments and histories
    • Ch3 Continued fractions
    • Ch4 Computation of the coefficients b(k) lambda(k)
    • Ch5 Orthogonality and exponential structures
    • Ch6 q-analogues
    • Lectures related to the course
    • Complements
    • References
  • Epilogue

The Art of Bijective Combinatorics    Part III
The cellular ansatz:  bijective combinatorics and quadratic algebra

The Institute of Mathematical Sciences, Chennai, India  (January-March 2018)

Some references

(list in progress)

PASEP  in  Physics

O. Angel, The stationary measure of a 2-type totally asymmetric exclusion process, J. Combin. Theory A, 113 (2006)  625-635, arXiv:math.PR/0501005
R.A. Blythe, M.R. Evans, F. Colaiori, F.H.L.Essler, Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra, J.Phys.A: math.Gen. 33 (2000) 2313-2332, arXiv:cond-mat/9910242.
R.A. Blythe, M.R. Evans, Nonequilibrium steady states of matrix product form: a solver’s guide, arXiv:  0706.1678 [cond-mat.stat.mec]  (127 pages)
R.A. Blythe, W. Janke, D.A. Johnson and R.Kenna, Continued fractions and the partially asymmetric exclusion process, arXiv 0904.3947 [cond-mat.stat.mec]
B. Derrida, M.R. Evans, V. Hakim and V. Pasquier, Exact solution of a one dimensional aysmmetric exclusion model using a matrix formulation, J. Phys. A: Math., 26 (1993) 1493-1517. 
B. Derrida, An exactly soluble non-equilibrium system: the asymmetric simple exclusion process, Physics Reports 301 (1998) 65-83, Proc. of the 1997-Altenberg Summer School on Fundamental problems in statistical mechanics. 
 B. Derrida, Matrix ansatz and large deviations of the density in exclusion process, invited conference, Proceedings of the International Congress of Mathematicians, Madrid, 2006. 
O.Golinelli, K.Mallick, Family of commuting operators for the totally asymmetric exclusion process, J.Phys.A:Math.Theor. 40 (2007) 5795-5812, arXiv: cond-mat/0612351
O.Golinelli, K.Mallick, The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics, J. Phys.A: Math.Gen.  39 (2006), arXiv:cond-mat/0611701

Combinatorics. TASEP

L.W. Shapiro and D. Zeilberger, A Markov chain occuring in enzyme kinetics, J. Math. Biology, 15 (1982) 351-357. 
E. Duchi and G. Schaeffer, A combinatorial approach to jumping particles, J. Combinatorial Th. A, 110 (2005) 1-29. 

Combinatorics. PASEP

J. Françon and X.G.Viennot  Permutations selon les pics, creux, doubles montées et doubles descentes, nombres d’Euler et nombres de Genocchi, Discrete Maths., 28 (1979) 21-35.
R. Brak and J.W. Essam, Asymmetric exclusion model and weighted lattice paths, J. Phys.A: Math Gen., 37 (2004) 4183-4217. 
S. Corteel, R. Brak, A. Rechnitzer and J. Essam, A combinatorial derivation of the PASEP stationary state, FPSAC’05, Taormina, 2005.
S. Corteel, A simple bijection between permutations tableaux and permutations, arXiv: math0609700
R. Brak, S.Corteel, J. Essam, R. Parviainen and A.Rechnitzer,. A combinatorial derivation of the PASEP stationary state. Electron. J. Combin. 13 (2006), no. 1, Research Paper 108, 23 pp.
S. Corteel and L.K. Williams, Tableaux combinatorics for the asymmetric exclusion process, Adv in Apl Maths, to appear, arXiv:math/0602109
S. Corteel and L.K Williams, A Markov chain on permutations which projects to the PASEP. Int. Math. Res. Not. (2007) article ID rnm055, arXiv:math/0609188
S. Corteel and  P.Nadeau, Bijections for permutation tableaux, Europ. J. of Combinatorics, 30 (2009), no. 1, 295–310.
 S. Corteel and L. K. Williams,  A Markov chain on permutations which projects to the PASEP. Int. Math. Res. Not. IMRN 2007, no. 17, Art. ID rnm055, 27 pp.
S. Corteel and L. K. Williams,  Tableaux combinatorics for the asymmetric exclusion process. Adv. in Appl. Math. 39 (2007), no. 3, 293–310.
S. Corteel and P. Nadeau,  Bijections for permutation tableaux.20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 13–24, Discrete Math. and Theor. Comput. Sci.
M. Josuat-Vergès, Rook placements in Young diagrams,  SLC 61
X.Viennot, Alternative tableaux, permutations and partially asymmetric exclusion process,   slides  (pdf, 10,4 Mo)         video
Workshop “Statistical Mechanics & Quantum-Field Theory Methods in Combinatorial Enumeration”,  
Isaac Newton Institute for Mathematical Science, Cambridge, 23 April 2008,   
J.C. Novelli, J.Y.Thibon and L.Williams, Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions ans permutations, arXiv:0804.0995
S. Corteel, M.Josuat-Vergès, T. Prellberg and M. Rubey, Matrix Ansatz, lattice path and rook placement, arXiv: 0811.4606  [math.CO]
M. Josuat-Vergès, Combinatorics of the three-parameter PASEP partition function, Electron. J. Combin. 18 (2011), Art. #P22.  arXiv 0912.1279   [math.CO]
M. Josuat-Vergès,  Generalized Dumont-Foata polynomials and alternative tableaux, SLC  (Séminaire Lotharingien de Combinatoire), B64b
P. Nadeau, The structure of alternative tableaux, arXiv: 0908.4050   [math.CO]

S. Corteel and S.Dasse-Hartaut, A bijection from staircase tableaux to inversion tables, giving some Eulerian and Mahonian statistics. Ann. Comb. 20 (2016), no. 4, 737–754.

Tree-like tableaux

J.-C. Aval, A. Boussicault, P. Nadeau, Tree-like tableaux FPSAC'2011, Reykjavik, Islande, DMTCS Proceedings (2011). 
J.-C. Aval, A. Boussicault, S. Dasse-Hartaut, The tree structure in staircase tableaux GASCom'2012, Bordeaux (2012). 
J.-C. Aval, A. Boussicault, M. Bouvel, M. Silimbani, Combinatorics of non-ambiguous trees FPSAC'2013, Paris (France), in DMTCS Proceedings FPSAC'2013, 49-60 (2013). 
J.-C. Aval, A. Boussicault, P. Nadeau, Tree-like tableaux, Elec. J. Comb. 20 (2013) P34. 
 J.-C. Aval, A. Boussicault, S. Dasse-Hartaut, Dyck tableaux, Theor. Comput. Sci. 502 (2013), 195-209. 
J.-C. Aval, A. Boussicault, M. Bouvel, M. Silimbani, Combinatorics of non-ambiguous trees Adv. Appl. Math., 56 (2014), 78-108. arXiv: 1305.3716 [math.CO]
J.-C. Aval, A. Boussicault, S. Dasse-Hartaut, The tree structure in staircase tableaux Sem. Lothar. de Comb., 70 (2014), B70g. 
E.Y.Jin, Heaps and exponential structures, arXiv 1407.0242v3  [math.CO]
J.-C. Aval, A. Boussicault, B. Delcroix-Oger, F. Hivert, P. Laborde- Zubieta, Non-ambiguous trees : new results ans generalizations, FPSAC'2016, Vancouver (Canada), July2016, arXiv: 1511.09455

Permutation tableaux

A. Burstein, On some properties of permutation tableaux, PP’06, June 2006, Reykjavik, Iceland. 
Postnikov, Total positivity, Grassmannians, and networks, arXiv: math.CO/0609764. 
E.Steingrimsson and L. Williams Permutation tableaux and permutation patterns,  J. Combinatorial Th. A., 114 (2007) 211-234. arXiv:math.CO/0507149
S. Corteel and J. S. Kim, Combinatorics on permutation tableaux of type A and type B. European J. Combin. 32 (2011), no. 4, 563–579.
 European J. Combin., 32 (2011), 563-579, arXiv

Catalan tableaux

X.G.Viennot, Catalan tableaux and the asymmetric exclusion process, in Proc. FPSAC’07 (Formal Power Series and Algebraic Combinatorics), Tienjin, Chine, 2007, 12 pp.   
J.-C. Aval, X. Viennot The product of trees in the Loday-Ronco algebra through Catalan alternative tableaux Semin. Lothar. de Combin. 63 (2010), B63h. 
J.-C. Aval, J.-C. Novelli J.-Y. Thibon The # product in combinatorial Hopf algebras FPSAC'2011, Reykjavik, Islande, DMTCS Proceedings (2011). 

Statistics on PASEP tableaux (permutation, alternative, tree-like and staircase tableaux)

S. Corteel and P. Hitczenko, Expected values of statistics on permutation tableaux. In 2007 Conference on Analysis of Algorithms, AofA 07, Discrete Math. Theor. Comput. Sci. Proc., AH, pages 325–339,  Nancy, 2007.

P. Hitczenko and S. Janson, Asymptotic normality of statistics on permutation tableaux. In Algorithmic probability and combinatorics, volume 520 of Contemp. Math., pages 83–104. Amer. Math. Soc., Providence, RI, 2010.  arXiv: 0904.1222  [math.CO]

S.Dasse-Hartaut and P. Hitczenko, greek letters in random staircase tableaux, Random Structures Algorithms, 42(1):73–96, 2013  arXiv: 1202.3092  [math.CO]

S. Corteel and S.Dasse-Hartaut, Statistics on staircase tableaux, Eulerian and Mahonian statistics. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 245–255, Discrete Math. Theor. Comput. Sci. Proc.
P. Hitczenko and S. Janson, Weighted random staircase tableaux, Combin. Probab. Comput., 23 (2014), 1114–1147.arXiv: 1212.5498  [math.CO]
P. Hitczenko and A. Parshall, On the distribution of parameters in random weighted staircase tableaux, Discrete Mathematics and Theoretical Computer Science, proc. BA, 2014,
157–168,  arXiv: 1404.3446  [math.CO]
P. Hitczenko and A. Lohss, On the asymptotic distribution of parameters in random weighted staircse tableaux, arXiv: 1509.05752 [math.CO]

P. Laborde-Zubieta. Occupied corners in tree-like tableaux. Sém. Lothar. Combin., 74: Art. B74b, 14, [2015-2017].  arXiv:1505.06098
P. Hitczenko and A. Lohss, Corners in tree-like tableaux, Proceedings of the 27th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorihtms, Krak'ow, Poland, 4-8 July 2016, and Electron. J. Combin., 23 (4) 2016.  arXiv: 1511.04989 [math.CO]
A. Lohss, The asymptotic distribution of symbols diagonals of random weightd staircase tableaux,  arXiv: 1512.00329 [math.CO]
A. L. L. Gao, E. X. L. Gao, P. Laborde-Zubieta, and B. Y. Sun. Enumeration of corners in tree-like tableaux. Discrete Math. Theor. Comput. Sci., 18(3):Paper No. 17, 26, 2016.
P. Hitczenko and A. Lohss, Probabilistic consequences of some polynomials recurrences, Proceedings of the 27th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorihtms, Krak'ow, Poland, 4-8 July 2016, arXiv: 1604.03164 [math.CO]
P. Hitczenko and Aleksandr Yaroslavskiy, Distribution of corners in tree-like tableaux, arXiv: 1804.02796 [math.CO]
R. Althoff, D. Diethrich, A. Lohss, Xin-Dee Low and E. Wichert, Expected value of statistics type-B permutation tableaux, arXiv: 2102.03225 [math.CO]

PASEP and orthogonal polynomials

T. Sasamoto, One-dimensional partially asymmetric simple exclusion process with open boundaries: orthogonal polynomials approach., J. Phys. A: math. gen. 32 (1999) 7109-7131
M.Uchiyama, T.Sassamoto, M.Wadati, Asymmetric simple exclusion process with open boundaries and Askey-Wilson polynomials, J. PhysA:Math.Gen.37 (2004) 4985-5002, arXiv: cond-math/0312457
S.Corteel, M.Josuat-Vergès, T. Prellberg, M. Rubey, Matrix ansatz, lattice paths and rook placements. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 313–324, Discrete Math. Theor. Comput. Sci. Proc., AK, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2009.
M. Josuat-Vergès, Enumération de tableaux et de chemins, moments de polynômes orthogonaux, thèse Université Paris Sud, Orsay, 25 janvier 2010.  (126 pages)
S. Corteel, M.Josuat-Vergès and L.K Williams,  The matrix Ansatz, orthogonal polynomials and
permutations, Adv. in Appl. Math. 46 (2011), no. 1-4, 209–225.  arXiv: 1005.2696 [math.CO]
S. Corteel and L.K Williams, Staircase tableaux, the asymmetric exclusion process, and Askey-Wilson polynomials. Proc. Natl. Acad. Sci. USA 107 (2010), no. 15, 6726–6730.
S. Corteel and L.K Williams, Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials. Duke Math. J. 159 (2011), no. 3, 385–415. arXiv:0910.1858 [math.CO]
S. Corteel and L.K Williams,Erratum to “Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials” [ MR2831874]. Duke Math. J. 162 (2013), no. 15, 2987–2996.
S. Corteel, R. Stanley, D.Stanton and L.K. Williams. Formulae for Askey-Wilson moments and enumeration of staircase tableaux. Trans. Amer. Math. Soc. 364 (2012), no. 11, 6009–6037. arXiv: 1007.5174  [math.CO].
S. Corteel, J.S.Kim and D.Stanton, Moments of orthogonal polynomials and combinatorics. Recent trends in combinatorics, 545–578, IMA Vol. Math. Appl., 159, Springer, [Cham], 2016.
.S. Kim and D.Stanton,  Moments of Askey-Wilson polynomials
 (with Dennis Stanton), J. Combin. Theory Ser. A, 125 (2014), 113-145.
S. Kim and D.Stanton, The combinatorics of associated Laguerre polynomials
 (with Dennis Stanton), SIGMA 11 (2015), 039.
S. Kim and D.Stanton, Bootstrapping and Askey-Wilson polynomials, J. Math. Anal. Appl., 421 (2015), 501-520.

2-species PASEP

M. Uchiyama. “Two-species asymmetric simple exclusion process with open boundaries”. Chaos, Solitons & Fractals 35 (2008), pp. 398–407. DOI. 

S. Corteel and L.K Williams,, Macdonald-Koornwinder moments and the two-species exclusion process, Selecta Mathematicae, Accepted 2017.
O.Mandelshtam and X.Viennot, Tableaux Combinatorics of the two-species PASEP, arXiv 1506.01980. [math.CO], To be published in J.Comb. Theory A.
O.Mandelshtam and X.Viennot,  Rhombic alternative tableaux and assemblées of permutations, arXiv 1609.07638 [math.CO]
O.Mandelshtam and X.Viennot,  Rhombic alternative tableaux, assemblées of permutations and the PASEP, Proc FPSAC’16, Vancouver, DMTCS 2016, 815-826.
S.Corteel, O.Mandelshtam and L.Williams, Combinatorics of the two-species ASEP and Koornwinder moments, Adv. in Math (2017) 160-204.
S.Corteel and A.Nunge, 2-species exclusion processes and combinatorial algebras, FPSAC 2017 (contributed talk), Sem. Lotharingien Comb. 2017.