Pith. sign in

REVIEW 2 major objections 4 minor 91 references

Combinatorial mappings of exclusion processes

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This review identifies the steady-state weights of the asymmetric exclusion process family with exact combinatorial counts of paths and permutations, and derives a new determinant formula for the TASEP partition function.

desk verdict A genuinely useful review with one new verified determinant formula and one honestly-flagged unproven bijection; the main claims hold up but the general-q interpolation is conjectural. read the letter →

arxiv 1908.00942 v2 pith:PEQSGRBL submitted 2019-08-02 cond-mat.stat-mech

Anthony J. Wood , Richard A. Blythe , Martin R. Evans This is my paper · ORCID
classification cond-mat.stat-mech MSC 82B2305A1560K35
keywords exclusionprocessesASEPTASEPmatrixproductstatescombinatorialenumerationMotzkinpathspermutationstatisticsRényientropy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the stationary-state probabilities of the asymmetric simple exclusion process family—TASEP, PASEP, SSEP—are not just computable but naturally combinatorial. Through the matrix-product representation of these steady states, each configuration weight can be read as an enumeration: for the TASEP, the number of dominated lattice paths under a given boundary path; for the SSEP, the number of permutations with a specified set of raised entries; and for general q, weighted versions of these objects. This identification matters because physical observables such as density profiles, correlation functions, and Rényi entropies become counting problems, and known results in enumerative combinatorics translate directly into physics. The paper also derives a new closed-form determinant for the TASEP partition function at general boundary rates, Eqs. (78)–(80). A sympathetic reader should take the main thesis to be that the nonequilibrium steady state of the ASEP is exactly encoded by these classical combinatorial structures.

What carries the argument

The machinery is the matrix-product representation of the steady state: each configuration is assigned to an ordered product of matrices D, E over occupied and empty sites, with reduction relations DE = qED + D + E and boundary vector conditions. Using explicit ladder-operator representations, these matrix strings become generating functions for lattice paths, specifically bicoloured Motzkin paths, and via a mapping from the combinatorial literature they also become generating functions for permutations of N+1 integers. The path-dominance formulation, where a configuration maps to a path and its weight counts the paths beneath it, carries the enumeration: its closure under the same reduction relations is what proves the equality of matrix weight and combinatorial count. The new determinant formula for the TASEP partition function arises from applying a Hessenberg-determinant recursion to the staircase path representing (D+E)^N.

What would settle it

Run the algorithm of Section 5.5 on all decorated bicoloured Motzkin paths for a small system, say N=4, and compare against all 120 permutations of {0,1,2,3,4}: if two distinct decorated paths produce the same permutation, or some permutation is never produced, the claimed one-to-one mapping is false. A numerical falsifier would be to check that the q-weight generating function over all decorated paths matches the known α=β=1 PASEP partition function term-by-term in q for N up to, say, 6.

Watch

Extended reading notes

Core claim

The central claim is that the matrix-product stationary weights of exclusion processes admit exact combinatorial interpretations, and these interpretations are genuinely useful. For the totally asymmetric case the weight of a configuration equals the number of lattice paths dominated by the path traced by particles and holes; this is equivalent to counting bicoloured Motzkin paths and gives Catalan and Narayana numbers as partition-function components. For the symmetric case the weight equals the number of permutations of {0,...,N} in which a prescribed set of integers is raised, giving Eulerian numbers and factorials. The paper extends these to partial asymmetry via q-weighted permutations and weighted bicoloured Motzkin paths, and contributes a new result: the TASEP partition function for general α, β can be written as the determinant of an N×N Hessenberg matrix (78)–(80). It also proposes a decorated bijection between Motzkin paths and permutations that would interpolate between the TASEP and SSEP pictures for general q.

Load-bearing premise

The load-bearing premise is the asserted one-to-one correspondence between decorated bicoloured Motzkin paths and permutations in Section 5.5; the paper itself notes that a formal proof would be welcome, and if this bijection fails, the claimed interpolation between the TASEP and SSEP pictures for general q would not be established.

Editorial extensions

If this is right

  • For the TASEP with α=β=1, the partition function is the Catalan number C_{N+1}, and the total weight of configurations with P particles is the Narayana number T(N+1,P+1).
  • The density profile and arbitrary-order correlation functions of the SSEP with α=β=1 follow by elementary counting of permutations, recovering linear profiles and product-form correlations.
  • The new determinant expression (78)–(80) gives a closed-form generating function for the TASEP partition function at arbitrary boundary rates, equivalent to the known series expansion.
  • The Rényi entropy of order two for the TASEP maps to enumerating walks in the upper quadrant; the paper gives its generating function and the phase-dependent asymptotic scaling of the sum of squared weights.
  • If the proposed decorated-Motzkin-to-permutation bijection holds, the one-to-many chain from ASEP configurations through dominated paths to permutations interpolates continuously between TASEP (q=0) and SSEP (q=1) for α=β=1.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the bijection of Section 5.5 is established, it would supply a uniform combinatorial mechanism behind the whole ASEP family: one extended state space of permutations whose q-weighting degenerates to path counting at q=0, and one might expect q-Eulerian identities to emerge as sums over decorated paths.
  • The determinant form of the partition function suggests that other observables, such as configuration weights with fixed particle numbers or boundary-condition sums, may also have Hessenberg-determinant representations, with the recursion (57) as a computational shortcut.
  • The path-dominance picture could be used as a sampling tool: generating uniform dominated paths, for example through the two-row Markov chain described in the paper, yields a direct route to TASEP weights, and the same construction may extend to multispecies processes through their queueing representation.
  • The Rényi-entropy mapping to upper-quadrant walks suggests that higher-order λ sums, currently unsolved, might be approached by the same kernel-method techniques if the step-set symmetry persists in λ dimensions.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This topical review surveys combinatorial interpretations of the stationary weights of the TASEP, PASEP, and SSEP within the matrix-product formalism, unifying them under dominated paths, bicoloured Motzkin paths, and permutations. It collects known results from the combinatorics and statistical-physics literature, presents new results including a determinant formula for the TASEP partition function (Eqs. (78)-(80)), and proposes in Section 5.5 a mapping between decorated bicoloured Motzkin paths and permutations intended to interpolate between the TASEP and SSEP pictures as q varies. The appendices supply derivations: Appendix C proves the path-dominance reduction relation, Appendix E proves the permutation reduction relations, and Appendix F verifies the determinant partition function by showing that its generating function matches the known TASEP generating function. The authors state explicitly that a formal proof of the proposed Motzkin-to-permutation bijection is still missing.

Significance. If the determinant formula and the various combinatorial interpretations hold, the paper is a useful synthesis for statistical physicists: the path-dominance formalism gives an intuitive account of TASEP weights, the permutation mapping yields exact SSEP density profiles and correlations, and the determinant identity is a compact new expression for the TASEP partition function. The paper is careful in several places: the partition-function determinant is checked via generating functions in Appendix F, the reduction relations for path dominance and permutations are demonstrated in Appendices C and E, and no parameters are fitted, so the results are exact. The main limitation is that the proposed Motzkin-to-permutation bijection in Section 5.5 is not proved; as it stands it is a conjecture and should be presented as such rather than as an established mapping.

major comments (2)
  1. [§5.5, 'Mapping between Motzkin paths and permutations for α = β = 1 and general q'] The central claim of a one-to-one mapping between decorated bicoloured Motzkin paths and permutations is asserted, but no proof is supplied; the text itself states that 'a formal proof of the proposition that there is a one-to-one mapping between decorated Motzkin paths and permutations would be welcome.' The arguments given, namely equal total cardinalities in the q→1 limit and a plausibility argument about relative order, do not establish injectivity or surjectivity of the proposed algorithm. Since this mapping is the basis for the claimed interpolation between the TASEP and SSEP pictures shown in Figure 6 and described in the abstract, please either provide a proof or explicitly label the mapping as a conjecture throughout the paper, and adjust the abstract and the Figure 6 caption so that the claim is not stated as an established result.
  2. [§5.5, last paragraph before Section 5.5.1] The argument that the normalisation (N+1)! 'would then follow' that every permutation is represented by exactly one decorated path relies on the unproved injectivity of the algorithm: equal cardinalities only yield a bijection after injectivity has been established. If two decorated paths can map to the same permutation, then even though the total numbers match, some permutation could be missed. Please make this logical dependency explicit and, in the absence of a proof, avoid presenting the surjectivity conclusion as a consequence of the counting argument.
minor comments (4)
  1. [§6.3.1, Eq. (108)] The series expansion '1 + 2z + 7z² + 30z⁴ + 146z⁵ + 772z⁶...' appears to omit the z³ term; the coefficient 30 should presumably attach to z³.
  2. [§6.2, text after Eq. (93)] The word 'functiion' should be 'function'.
  3. [§7.1 and §7.2] There are typographical errors such as 'paricles' in Section 7.2; these should be corrected in a careful copy-edit.
  4. [Abstract and §2.3.3] The phrase 'one-to-many mapping' is used to describe the relation from ASEP configurations to paths or permutations, but the direction described in the text is one configuration mapping to many extended objects; consider using the clearer 'one-to-many' / 'many-to-one' terminology or explicitly define the direction of the maps in the text.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the weight mappings are proven by reduction-relation checks, the new determinant is independently checked, and the Section 5.5 bijection is an admitted proof gap rather than a circular step.

full rationale

The paper's central derivations are self-contained checks against the matrix-product reduction relations, not fits disguised as predictions. Section 3.1 defines W(T) as an enumeration of dominated paths and proves in Appendix C that W satisfies the same relations as the q=0, alpha=beta=1 matrix product (DE=D+E), so the equality W(C)=<W|...|V> is a proven representation, not an input. Similarly, Section 4.1 and Appendix E prove the permutation-counting weights satisfy the SSEP/PASEP reduction relations including the q-deformed DE=qED+D+E, so the permutation mapping is independently established. The new determinant formula (78)-(80) is obtained algebraically from Mandelshtam's external formula (75)-(76), and Appendix F verifies it by matching the known generating function for Z_N; this is a standard independent consistency check, and although the generating function is quoted from the authors' own review [9], the partition function has independent derivations [7,20] and is not fitted to the determinant. Section 5.5's decorated-Motzkin-to-permutation mapping is explicitly not proved: the paper says 'A formal proof of the proposition that there is a one-to-one mapping between decorated Motzkin paths and permutations would be welcome.' Equal cardinality plus a map does not by itself establish a bijection, so this is a load-bearing correctness gap for the claimed q-interpolation, but it is a missing proof, not a circular reduction: nothing is defined in terms of the conclusion, and no fitted parameter is renamed as a prediction. The self-citations [13,22,26,9] are used to import established calculations and are not invoked to forbid alternative derivations. Overall circularity is negligible.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central derivations rest on the established matrix product solution of the ASEP and on classical enumerative identities. The only newly introduced structural object is the decorated Motzkin path, whose bijective status is open. No parameters are fitted to data; alpha, beta and q are physical model parameters.

assumptions (4)
  • domain assumption The matrix product representation (Eqs. 2-5) solves the ASEP stationary state (from Derrida, Evans, Hakim and Pasquier 1993)
    The paper takes this as the starting point and does not reprove it; it is an established theorem in statistical physics.
  • standard math Mandelshtam's determinant formula for general alpha, beta TASEP weights (Eqs. 75-76)
    Imported from [37] and used to derive the new determinant form of the partition function; assumed correct without proof in this paper.
  • standard math Standard enumerative identities: Catalan, Narayana, Eulerian numbers, Chu-Vandermonde, reflection principle
    Used without proof to evaluate sums and to identify integer sequences; these are classical results in combinatorics.
  • domain assumption Explicit semi-infinite matrix representations (Eqs. 12-15 and 81-82) yield the correct ASEP weights in the appropriate limits
    The review relies on these representations, citing [9,13,16], to translate matrix products into path counts; they are not derived in this paper.
invented entities (1)
  • Decorated bicoloured Motzkin paths with baubles
    purpose: To encode q-dependent weights and to interpolate between the Motzkin path picture of the TASEP and the permutation picture of the SSEP (Section 5.5).
    Introduced in this paper; its summed weight matches the known weighted-Motzkin partition function for alpha=beta=1, but the claimed one-to-one mapping to permutations is unproven and no external falsifiable prediction is offered.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Combinatorial mappings of exclusion processes." pith.science (2026). https://pith.science/paper/PEQSGRBL

@misc{pith2026190800942,
  author       = {Pith},
  title        = {Pith review of: Combinatorial mappings of exclusion processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PEQSGRBL}},
  note         = {Machine review of arXiv:1908.00942}
}
read the original abstract

We review various combinatorial interpretations and mappings of stationary-state probabilities of the totally asymmetric, partially asymmetric and symmetric simple exclusion processes (TASEP, PASEP, SSEP respectively). In these steady states, the statistical weight of a configuration is determined from a matrix product, which can be written explicitly in terms of generalised ladder operators. This lends a natural association to the enumeration of random walks with certain properties. Specifically, there is a one-to-many mapping of steady-state configurations to a larger state space of discrete paths, which themselves map to an even larger state space of number permutations. It is often the case that the configuration weights in the extended space are of a relatively simple form (e.g., a Boltzmann-like distribution). Meanwhile, various physical properties of the nonequilibrium steady state - such as the entropy - can be interpreted in terms of how this larger state space has been partitioned. These mappings sometimes allow physical results to be derived very simply, and conversely the physical approach allows some new combinatorial problems to be solved. This work brings together results and observations scattered in the combinatorics and statistical physics literature, and also presents new results. The review is pitched at statistical physicists who, though not professional combinatorialists, are competent and enthusiastic amateurs.

Figures

Figures reproduced from arXiv: 1908.00942 by the authors.

Figure 1
Figure 1. The exclusion process that constitutes the bulk of this review. Particles attempt to enter the system at rate α, attempt to move to the right at rate 1, to the left at rate q, and exit from the rightmost site at rate β. Particles may only move into free sites. the matrix product approach may help solve otherwise challenging combinatorial problems [10]. We begin the review by discussing the the totally asymmetric, si… view at source ↗
Figure 2
Figure 2. Phase diagram of the TASEP and typical density profiles hτii in each of the three phases. explicit representation. One can go on to calculate physical observables, such as the current and density profile, in this way [7]. However, for the purposes of identifying mappings to combinatorial enumeration problems, it is often helpful to write out D, E, hW|, |V i explicitly. Generally, no finite-size matrices obey (2–5) (… view at source ↗
Figure 3
Figure 3. A bicoloured Motzkin path (left), and its equivalent Dyck path (right). scalar product with a bra vector hn|ki = δnk. From this, and on setting α = β = 1, the representations (12–15) simplify to D = 1 + g =   1 1 0 0 · · · 0 1 1 0 · · · 0 0 1 1 · · · 0 0 0 1 · · · . . . . . . . . . . . . . . .   E = 1 + g † =   1 0 0 0 · · · 1 1 0 0 · · · 0 1 1 0 · · · 0 0 1 1 · · · . . . . . . . . . . . . . . .… view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: Left: a Dyck path, consisting of equal numbers of up-steps and down-steps such that the path never goes below 0. Right: a walk that starts and ends at 0, but goes below, and its reflection about the point it first touches −1, which then terminates at −2. 2.3.3. The one…
Figure 5
Figure 5. Figure 5: Dyck paths and one-transit walks. The first row illustrates the three Dyck paths with 2N = 8 steps and p = 3 returns. The second row illustrates the four one￾transit walks corresponding to the first Dyck path: the Dyck path inverted at each return to make a set of p + …
Figure 6
Figure 6. Figure 6: Schematic of the combinatorial mappings to be outlined in this review. An ASEP configuration (an example is illustrated in the left column) has a one-to￾many mapping to certain dominated paths (illustrated in the middle column), which we propose in Section 5.5 to in tu…
Figure 7
Figure 7. Figure 7: Left: the path T with the three equivalent specifications (35), (36) and (37). Right: two paths T , T 0 . Here, T dominates T 0 . 3.1. Mapping to a path dominance problem Consider the set of discrete paths T ∈ {↑,→}N that begin at (0, 0), and end at (Q, P), with P + Q …
Figure 8
Figure 8. Figure 8: The weight of the path (↑, ↑, →, →) is 6, as 6 distinct paths can be drawn within its perimeter. This quantity can be written out iteratively, accumulating all possible dominated paths as T grows step by step. Formally, this is W(T ) = X y0 n0=0 X y1 n1=n0 X y2 n2=n1 ·…
Figure 9
Figure 9. Figure 9: Graphical representation of (44), (45). Adding a → to the start or a ↑ to the end of a path does not change its weight (i.e., the number of paths it can dominate). 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9…
Figure 10
Figure 10. Figure 10: Graphical representation of (46). are trivial by inspection ( [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: Calculation of the TASEP partition function for N = 3. For each configuration (left), we draw draw all combinations of length-N paths that dominate another (centre), and their equivalent bicoloured Motzkin path (right). The particles may be arranged in any way across …
Figure 12
Figure 12. Figure 12: Simplified two-row dynamics for the case α = β = 1 which generates a uniform distribution over the space of complete configurations. Particles hop clockwise into empty spaces around the lattice at unit rate. Vertical dashed lines indicate zone boundaries, wherein each…
Figure 13
Figure 13. Figure 13: Example of a complete configuration in [39] (left) and its equivalent Motzkin path (centre) and dominated path (right). The top row of the complete configuration shows that these correspond to C = (1, 0, 1, 0, 0, 0, 1). satisfy a generalisation of detailed balance (se…
Figure 14
Figure 14. Figure 14: Partition function (D + E) N expressed as a ‘staircase’ path. the partition function is the weight of a single ‘staircase’ path of length 2N (see [PITH_FULL_IMAGE:figures/full_fig_p028_14.png]
Figure 15
Figure 15. Figure 15: The weight of the path T = (↑, →, →, ↑), corresponding to the TASEP configuration C = (1, 0, 0, 1). Both have weight W(T ) = 1/α2β + 1/αβ + 1/β2 = hW|DEED|V i [PITH_FULL_IMAGE:figures/full_fig_p029_15.png]
Figure 16
Figure 16. Figure 16: Weight of the Motzkin path (%, ·, %, &, &, ×). This is one of many paths mapping to the configuration C = (1, 1, 1, 0, 0, 0). Note that this representation is distinct from (12–15). This representation lends a natural association of weights on the bicoloured Motzkin p…
Figure 17
Figure 17. Figure 17: A decorated bicoloured Motzkin path of length N = 6, with baubles (red, starred) at heights mi = (0, 0, 1, 1, 0, 0, 0). The weight of this path is q 2 . Each possible decorated bicoloured Motzkin path can be translated into a permutation of the integers (0, 1, . . . N…
Figure 18
Figure 18. Figure 18: Example of a 2D walk comprising the steps {↑, ↓, →, ←, ., %} from (i, j) = (1, 1) to (k, l) = (7, 2). The walk must remain in the upper quadrant, but may touch and move along the boundary. defining the ladder operators g1, g2: g1|ki ⊗ |li = |k − 1i ⊗ |li , g † 1 |ki ⊗…
Figure 19
Figure 19. Figure 19: Graphical representation of the squared weight of a path in the path dominance formalism. weights emerges, with a different scaling for the three phases. After normalising, X C P(C) 2 ∼    f(α, β) [PITH_FULL_IMAGE:figures/full_fig_p037_19.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

91 extracted references · 78 canonical work pages

  1. [1]

    Schadschneider A, Chowdhury D and Nishinari K 2010 Stochastic transport in complex systems: from molecules to vehicles (Elsevier)

    2010

  2. [2]

    Chou T, Mallick K and Zia R K P 2011 Reports on Progress in Physics 74 116601

    2011

  3. [3]

    Lieb E H and Mattis D C 2013 Mathematical physics in one dimension: exactly soluble models of interacting particles (Academic Press)

    2013

  4. [4]

    Evans M R 2000 Brazilian Journal of Physics 30 42–57

    2000

  5. [5]

    Derrida B, Domany E and Mukamel D 1992 J. Stat. Phys. 69 667

    1992

  6. [6]

    Derrida B and Evans M R 1993 Journal de Physique I 3 311–322

    1993

  7. [7]

    Derrida B, Evans M R, Hakim V and Pasquier V 1993 Journal of Physics A: Mathematical and General 26 1493

    1993

  8. [8]

    Sch¨ utz G and Domany E 1993J. Stat. Phys. 72

Show all 91 references
  1. [9]

    Blythe R A and Evans M R 2007 J. Phys. A: Math. Theor. 40 R333

    2007

  2. [10]

    Corteel S, Josuat-Verg` es M and Williams L K 2011Advances in Applied Mathematics 46 209–225

  3. [11]

    Uchiyama M, Sasamoto T and Wadati M 2004 J. Phys. A.: Math. Gen. 37 4985

    2004

  4. [12]

    Uchiyama M 2008 Chaos Solitons Fractals 35 398

    2008

  5. [13]

    Blythe R, Evans M, Colaiori F and Essler F 2000 Journal of Physics A: Mathematical and General 33 2313

    2000

  6. [14]

    Essler F H L and Rittenberg V 1996 Journal of Physics A: Mathematical and General 29 3375– 3407

    1996

  7. [15]

    Mallick K and Sandow S 1997 Journal of Physics A: Mathematical and General 30 4513–4526

    1997

  8. [16]

    Sasamoto T 1999 J. Phys. A.: Math. Gen. 32 7109

    1999

  9. [17]

    Derrida B, Lebowitz J and Speer E 2002 Journal of statistical physics 107 599–634

    2002

  10. [18]

    Sasamoto T, Mori S and Wadati M 1996 Journal of the Physical Society of Japan 65 2000–2008

    1996

  11. [19]

    Vanicat M 2017 Journal of Statistical Physics 166 1129–1150

    2017

  12. [20]

    thesis University of Oxford

    Depken M 2003 Models of non-equilibrium systems Ph.D. thesis University of Oxford

    2003

  13. [21]

    Blythe R, Janke W, Johnston D and Kenna R 2004 Journal of Statistical Mechanics: Theory and Experiment 2004 P06001

    2004

  14. [22]

    Wood A J, Blythe R A and Evans M R 2017 J. Phys. A: Math. Theor. 50 475005

    2017

  15. [23]

    Gould H W 1956 The American Mathematical Monthly 63 84–91

    1956

  16. [24]

    Stanley R P and Fomin S 1999 Enumerative Combinatorics (Cambridge Studies in Advanced Mathematics vol 2) (Cambridge University Press)

    1999

  17. [25]

    Brak R, de Gier J and Rittenberg V 2004 J. Phys. A.: Math. Gen. 37 4303 Combinatorial mappings of exclusion processes 46

    2004

  18. [26]

    Blythe R A, Janke W, Johnston D A and Kenna R 2004 J. Stat. Mech.: Theor. Exp. P10007

    2004

  19. [27]

    Deutsch E 1999 Discrete Mathematics 204 167–202

    1999

  20. [28]

    Derrida B, Evans M and Mukamel D 1993 Journal of Physics A: Mathematical and General 26 4911

    1993

  21. [29]

    Comtet L 2012 Advanced Combinatorics: The art of finite and infinite expansions (Springer Science & Business Media)

    2012

  22. [30]

    Brak R and Essam J W 2001 Journal of Physics A: Mathematical and General 34 10763–10782

    2001

  23. [31]

    Kreweras G and Niederhausen H 1981 Eur. J. Comb. 2 55–60

    1981

  24. [32]

    Kreweras G 1965 Cahiers du Bureau universitaire de recherche op´ erationnelle S´ erie Recherche6 9–107

    1965

  25. [33]

    Numer 33 261–273

    Niederhausen H 1981 Congr. Numer 33 261–273

    1981

  26. [34]

    Narayana T 1955 Journal of the Indian Society of Agricultural Statistics 5 169–178

    1955

  27. [35]

    Sloane N J A 1996 The on-line encyclopedia of integer sequences, sequence A001263

    1996

  28. [36]

    Kaygisiz K and Sahin A 2013 Bulletin of the Iranian Mathematical Society 39 1065–1078

    2013

  29. [37]

    Mandelshtam O 2015 Journal of Combinatorial Theory, Series A 132 120–141

    2015

  30. [38]

    Brak R and Essam J 2004 Journal of Physics A: Mathematical and General 37 4183

    2004

  31. [39]

    Duchi E and Schaeffer G 2005 Journal of Combinatorial Theory, Series A 110 1–29

    2005

  32. [40]

    Kelly F P 1979 Reversibility and stochastic networks (Chichester: Wiley)

    1979

  33. [41]

    Corteel S and Williams L K 2007 International mathematics research notices 2007 rnm055–rnm055

    2007

  34. [42]

    Carinci G, Giardin` a C, Giberti C and Redig F 2013 Journal of Statistical Physics 152 657–697

    2013

  35. [43]

    Spohn H 1983 Journal of Physics A: Mathematical and General 16 4275

    1983

  36. [44]

    Derrida B, Dou¸ cot B and Roche P E 2004 Journal of Statistical physics 115 717–748

    2004

  37. [45]

    Worpitzky J 1883 Journal f¨ ur die reine und angewandte Mathematik 94 203–232

  38. [46]

    Carlitz L 1959 Mathematics Magazine 32 247–260

    1959

  39. [47]

    Sloane N J A 1996 The on-line encyclopedia of integer sequences, sequence A008292

    1996

  40. [48]

    Petersen T K 2015 Eulerian numbers Eulerian Numbers (Springer) pp 3–18

    2015

  41. [49]

    Carlitz L 1954 Transactions of the American Mathematical Society 76 332–350

    1954

  42. [50]

    Corteel S and Williams L K 2007 Advances in applied mathematics 39 293–310

    2007

  43. [51]

    Williams L K 2005 Advances in Mathematics 190 319–342

    2005

  44. [52]

    Brak R, Corteel S, Essam J, Parviainen R and Rechnitzer A 2006 the electronic journal of combinatorics 13 108

    2006

  45. [53]

    Blythe R A, Janke W, Johnston D A and Kenna R 2009 J. Phys. A: Math. Theor. 42 325002

    2009

  46. [54]

    Corteel S and Williams L K 2011 Duke Mathematical Journal 159 385–415

    2011

  47. [55]

    Corteel S, Stanley R, Stanton D and Williams L 2012 Transactions of the American Mathematical Society 364 6009–6037

    2012

  48. [56]

    Josuat-Verg` es M 2011Electron. J. Comb. 18 P22

  49. [57]

    R´ enyi A 1961 On measures of entropy and information Proceedings of the fourth Berkeley symposium on mathematical statistics and probability vol 1 pp 547–561

    1961

  50. [58]

    Baez J C 2011 Renyi Entropy and Free Energy arXiv:1102.2098

    2011 arXiv

  51. [59]

    Math 520 1–40

    Bousquet-M´ elou M and Mishna M 2010Contemp. Math 520 1–40

  52. [60]

    Sloane N J A 1996 The on-line encyclopedia of integer sequences, sequence A196148

    1996

  53. [61]

    Sloane N J A 1996 The on-line encyclopedia of integer sequences, sequence A111910

    1996

  54. [62]

    Bostan A, Bousquet-M´ elou M, Kauers M and Melczer S 2016Annals of Combinatorics 20 661–704

  55. [63]

    Bacher A, Kauers M and Yatchak R 2015 arXiv preprint arXiv:1511.05763

    2015 arXiv

  56. [64]

    Derrida B, Janowsky S A, Lebowitz J L and Speer E R 1993 Journal of Statistical Physics 73 813–842

    1993

  57. [65]

    Evans M R, Foster D P, Godr` eche C and Mukamel D 1995 J. Stat. Phys. 80 69–102

    1995

  58. [66]

    Arita C 2006 J. Stat. Mech.: Theor. Exp. 2006 P12008–P12008

    2006

  59. [67]

    Ayyer A, Lebowitz J L and Speer E R 2009 Journal of Statistical Physics 135 1009–1037

    2009

  60. [68]

    Cantini L 2017 Ann. Henri. Poincar´ e18 1121

    2017

  61. [69]

    Aas E, Ayyer A, Linusson S and Potka S 2019 The Exact Phase Diagram For A Semipermeable Combinatorial mappings of exclusion processes 47 Tasep With Nonlocal Boundary Jumps arXiv:1902.02019

    2019 arXiv

  62. [70]

    Crampe N, Mallick K, Ragoucy E and Vanicat M 2015 J. Phys. A: Math. Theor. 48 175002

    2015

  63. [71]

    Crampe N, Evans M R, Mallick K, Ragoucy E and Vanicat M 2016 J. Phys. A: Math. Theor. 49 475001

    2016

  64. [72]

    Crampe N, Ragoucy E and Vanicat M 2014 Journal of Statistical Mechanics: Theory and Experiment 2014 P11032

    2014

  65. [73]

    Ferrari P A, Fontes L R G and Kohayakawa Y 1994 Journal of Statistical Physics 76 1153–1177

    1994

  66. [74]

    Angel O 2006 Journal of Combinatorial Theory, Series A 113 625 – 635

    2006

  67. [75]

    Ferrari P A and Martin J B 2007 Ann. Probab. 35 807–832

    2007

  68. [76]

    Evans M R, Ferrari P A and Mallick K 2009 Journal of Statistical Physics 135 217–239

    2009

  69. [77]

    Martin J 2018 Stationary distributions of the multi-type ASEP arXiv:1810.10650

    2018 arXiv

  70. [78]

    Arita C and Mallick K 2013 J. Phys. A.: Math. Theor. 46 085002

    2013

  71. [79]

    Ayyer A and Linusson S 2014 Adv. Appl. Math. 57 21

    2014

  72. [80]

    Prolhac S, Evans M R and Mallick K 2009 J. Phys. A: Math. Theor. 42 165004

    2009

  73. [81]

    Finn C, Ragoucy E and Vanicat M 2018 Journal of Statistical Mechanics: Theory and Experiment 2018 043201

    2018

  74. [82]

    Arita C, Ayyer A, Mallick K and Prolhac S 2011 J. Phys. A: Math. Theor. 44 335004

    2011

  75. [83]

    Arita C, Ayyer A, Mallick K and Prolhac S 2012 J. Phys. A: Math. Theor. 45 195001

    2012

  76. [84]

    Kuniba A, Maruyama S and Okado M 2015 J. Phys. A: Math. Theor. 48 34FT02

    2015

  77. [85]

    Kuniba A, Maruyama S and Okado M 2016 J. Phys. A: Math. Theor. 49 114001

    2016

  78. [86]

    Cantini L, de Gier J and Wheeler M 2015 J. Phys. A: Math. Theor. 48 334001

    2015

  79. [87]

    Cantini L, Garbali A, de Gier J and Wheeler M 2016 J. Phys. A: Math. Theor. 49 444002

    2016

  80. [88]

    Corteel S, Mandelshtam O and William L 2019 arXiv preprint arXiv:1811.01024

    2019 arXiv

  81. [89]

    Harary F 1969 Graph Theory (Addison-Wesley)

    1969

  82. [90]

    Schnakenberg J 1976 Reviews of Modern physics 48 571

    1976

  83. [91]

    Exact expression for ASEP partition function Here we present a general expression for the PASEP partition function derived in [13] and its specialisation to the case α =β = 1

    Askey R 1975 Orthogonal polynomials and special functions vol 21 (Siam) Appendix A. Exact expression for ASEP partition function Here we present a general expression for the PASEP partition function derived in [13] and its specialisation to the case α =β = 1. ZN = ( 1 1−q )N N...

    1975

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.