{"id":"c98fa296-c2ab-41df-b9fe-ae51a587bf9b","arxiv_id":"2507.23460","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Fuss-Catalan algebras and their one- and two-boundary versions are realized on increasing chains of non-crossing partitions, with a new r=2 reflection equation solution.","lead":"This paper unifies the combinatorics of Fuss-Catalan algebras, a known generalization of the Temperley-Lieb algebra, using non-crossing partitions and generalized Dyck paths, and reports a new boundary solution of the reflection equation for the r=2 case. A generalist reader will find a data-free, diagram-based way to see why several different algebraic pictures encode the same counting problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.8's stacking argument for chains of length >2 is asserted, not proved; the central isomorphism of Proposition 5.5 depends on it.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing gap: the stacking argument in Proposition 4.8 for chains longer than a single cover. I agree that this is the most critical point because Proposition 5.5, the central isomorphism between the chain-based Fuss–Catalan algebra NC(r)_n and the diagrammatic algebra TL(r)_n, is proved by exhibiting Ψ(r) as an intertwining bijection. If Ψ(r) is not well-defined for all increasing r-chains, the map F(s)_i ↦ E(s)_i does not even define an action, and the main combinatorial claim fails. The boundary versions in Theorems 7.10 and 8.12 inherit the same dependence, since they are built on symmetric analogues of the same bijection. The proof in §4.4 is not merely terse; it skips a nontrivial transitivity property. Pairwise non-crossing for adjacent covers does not by itself imply non-crossing for the endpoints of a longer chain, because each cover reconnects arches and could create a crossing with an earlier partition's chord diagram. The paper's assertion that 'repeating a similar argument' suffices is exactly the kind of step that needs a rigorous induction or a computational sanity check. I do not see an internal inconsistency in the r=1 case: Proposition 3.11 and the bijection Ψ are worked out in detail and are standard in spirit. The r≥2 construction is plausible and likely correct, but the missing proof is a genuine correctness risk, not a matter of outside consensus. Hence the reader's CONDITIONAL verdict is appropriate. I would not escalate to REJECT because the gap is localized and testable: a small exhaustive computation could quickly either find a counterexample or support the claim. The reflection-equation section has its own issues (e.g., the apparent z/w typo around Eqs. (9.17)–(9.18) and the asserted uniqueness of C1, C2), but those are secondary to the main combinatorial isomorphism and do not affect the central claim of the paper. Therefore my stress-test pass leaves the reader's verdict unchanged.","tokens_in":49220,"tokens_out":5997,"duration_ms":59653,"concrete_test":"Enumerate all increasing r-chains for small n (e.g., n ≤ 6, r ≤ 4) and computationally check: (1) for every chain, the superposition of Ψ(π_1),...,Ψ(π_r) is a non-crossing chord diagram satisfying (2.1); (2) for every C ∈ C(r)_n, the inverse construction yields an increasing chain. Also test the transitivity property directly: for all π ≤ ν ≤ ρ in NC_n, if Ψ(π) and Ψ(ν) are non-crossing and Ψ(ν) and Ψ(ρ) are non-crossing, then Ψ(π) and Ψ(ρ) are non-crossing. A single counterexample would falsify Prop 4.8; if all small cases pass, the missing rigorous induction remains, but the claim is not refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central isomorphism of Proposition 5.5 rests on the bijection Ψ(r) of §4.4, whose well-definedness is Proposition 4.8. The proof of 4.8 treats only a single cover π1 ⋖ π2 in two block configurations, then asserts that for an arbitrary chain π1 ≤ π2 one can 'repeat a similar argument' along the cover sequence (4.2). This requires a transitivity statement that is not established: if (Ψ(π),Ψ(ν)) and (Ψ(ν),Ψ(ρ)) are non-crossing in C(2)_n for π ≤ ν ≤ ρ, it does not automatically follow that (Ψ(π),Ψ(ρ)) is non-crossing, since the cover operations reconnect endpoints and could create a crossing with an earlier chord diagram. The subsequent step to r > 2 ('Since π_i ≤ π_j ... it is easy to see') inherits this gap. Moreover, the inverse direction of Ψ(r) is only sketched: given C ∈ C(r)_n, the partitions π_s = Ψ^{-1}(C_s) are asserted to form an increasing r-chain without proof. If the stacking or the inverse chain condition fails for some longer chain, Ψ(r) is not a bijection and the isomorphism F(s)_i ↔ E(s)_i collapses. The boundary analogues (Theorems 7.10, 8.12) rely on the same Ψ(r) for symmetric chains, so the concern propagates.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a combinatorial framework for Fuss–Catalan algebras using increasing r-chains of non-crossing partitions. It defines a bijection between such chains and generalized Dyck paths, a second bijection to generalized chord diagrams, and uses these to compare an algebra generated by operators on chains with the diagrammatic Fuss–Catalan algebra of Bisch–Jones and Di Francesco. Boundary analogues are introduced via symmetric non-crossing partitions and primed integers, and a solution of the reflection equation for r = 2 is proposed. The central claim is that the chain algebra is isomorphic to the diagrammatic algebra, with the boundary versions isomorphic to the one- and two-boundary diagrammatic algebras.","tokens_in":49328,"tokens_out":6159,"duration_ms":74026,"significance":"If the missing verifications are supplied, the paper would give a useful and explicit dictionary between increasing chains of non-crossing partitions, generalized Dyck paths, and generalized chord diagrams, extending the classical Temperley–Lieb/non-crossing partition correspondence. The paper is constructive: the bijections are described algorithmically, several examples are worked out, and the r = 1 case is checked against the known Temperley–Lieb action. The proposed r = 2 reflection matrix is an explicit, falsifiable formula. However, several load-bearing assertions are supported only by sketches or by 'repeating a similar argument', so the announced isomorphisms are not yet fully established.","major_comments":[{"comment":"The proof of Proposition 4.8 establishes only the single-cover case π1 ⋖ π2 and then extends to arbitrary chains by 'repeating a similar argument' along the composition sequence (4.2). This requires a transitivity statement that is not proved: if the pairs (Ψ(π1), Ψ(ν)) and (Ψ(ν), Ψ(π2)) are non-crossing, it does not follow automatically that (Ψ(π1), Ψ(π2)) is non-crossing, since each cover operation reconnects endpoints and can create new chords that were not present in the earlier diagram. The inverse direction of Ψ(r) is also only sketched: after reading off the chord diagrams Cs from C, the partitions πs = Ψ^{-1}(Cs) are asserted to form an increasing chain without proof. Since Proposition 5.5 and the boundary analogues in Theorems 7.10 and 8.12 all pass through Ψ(r), this gap is load-bearing.","section":"§4.4, Proposition 4.8"},{"comment":"Even granting Proposition 4.8, the proof of Proposition 5.5 asserts that F_i^(s) = Ψ(r)^{-1} E_i^(s) Ψ(r) 'by the construction of the bijection Ψ(r)'. The reader is not shown why applying f1 to the suffix π_{r-s+1},...,π_r of an r-chain corresponds, under the superposition Ψ(r), to the action of E_i^(s) on exactly the s inner strands of the generalized chord diagram. This requires a diagrammatic comparison that is not supplied, and the same gap propagates to Theorem 7.10 and Theorem 8.12, whose proofs reduce to the r = 1 case plus a superposition assertion.","section":"§5, Proposition 5.5"},{"comment":"The bijection SN C'^{(r)}_n ≅ SC'^{(r)}_n is asserted by comparing Definition 6.23 with Definition 8.10, but the constraints in Definition 6.23(a)–(b) and the admissibility condition in Definition 8.10 involve both the chain order and the nesting order of symmetric chords. No proof is given that these constraints correspond under the bijection of Proposition 6.7, nor that the generators G_i^(s) preserve the set SN C'^{(r)}_n. This is load-bearing for the two-boundary isomorphism Theorem 8.12.","section":"§8, Lemma 8.13 and Definition 8.10"}],"minor_comments":[{"comment":"The title header contains the typo 'FUSS–CA T ALAN'; the introduction also has 'Temperely' for 'Temperley'.","section":"Title and running header"},{"comment":"Example 4.6 says 'an increase 3-chain' where 'an increasing 3-chain' is meant, and the phrase 'the positions of U in P_{j1∪...∪jm} is given by' should agree in number with 'set'.","section":"Example 4.6"},{"comment":"In the merging step, the text writes 'a := {q ∈ B^{(i-1)}_{j1} : q < min B^{(i-1)}_{j2}}' and then uses 'a(r+1)' as a position; it should specify that the cardinality |a| is intended.","section":"§4.2, construction of κ(r)"},{"comment":"The chain of equalities ξ^{r+1} = eσ^{r+1} = ρ^2 mixes maps on three different sets; the statement should be clarified by explicitly naming the identifications under which the equality is asserted.","section":"Proposition 4.7"},{"comment":"The proof says the relations are 'routine to check' and lists only relations of order up to three; since the paper later uses higher-order diagram identities, it would be helpful to state explicitly that all relations are determined by the diagrammatic representation and to give one worked higher-order example.","section":"Theorem 5.3"},{"comment":"Reference [27] contains the typo 'J. Pure Appl. Alegebra' and should be corrected.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central construction is plausible and the gaps appear fixable by adding a careful proof of Proposition 4.8, including the inverse direction, and by verifying that the boundary parity/admissibility conditions are preserved under the actions. The paper does not need to be rejected on the basis of the current gaps, but the announced isomorphisms should not be accepted as proved until these points are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a substantial, mostly checkable paper that builds a uniform non-crossing partition model for the Temperley-Lieb, Fuss-Catalan, and boundary Fuss-Catalan algebras, and it closes with a claimed new r=2 reflection solution. I would not desk-reject it, but the referee should ask for a real proof of the stacking lemma before the central isomorphism is trusted.\n\nThe genuinely new material is the increasing r-chain framework: the bijections kappa(r), Psi(r), and Phi, the cover-exclusive Dyck tiling description, the boundary algebras built from symmetric non-crossing partitions with primed integers, and the r=2 reflection solution. The r=1 part is mostly a clean repackaging of known compatibility between TL actions and Kreweras rotation, but it is explicit and the commutative-diagram checks (Fi = Psi^{-1} ei Psi, rho = Psi^{-1} sigma Psi) are verifiable. The examples throughout are useful, and the paper engages properly with the literature: Edelman, Stump, Bisch-Jones, Di Francesco, and de Gier-Nichols are all used, not just cited.\n\nThe main soft spot is exactly the one in the stress-test note. Proposition 4.8 is load-bearing: it asserts that the chord diagrams coming from an increasing chain can be superposed into a generalized chord diagram in C(r)_n. The proof treats a single cover in two block configurations and then says \"repeating a similar argument\" for longer chains. That is a real gap, because the cover operations reconnect endpoints and the non-crossing property is not obviously transitive across a chain. The inverse direction of Psi(r) is also only sketched: given C in C(r)_n, the partitions Psi^{-1}(C_s) are asserted to form an increasing chain without proof. Propositions 5.5, 7.10, and 8.12 all sit on this, so the gap propagates. I think the claim is probably true, but it is not yet demonstrated.\n\nTheorem 5.3 says the diagram relations are \"routine to check\"; that is acceptable for a diagram algebra, but it does not cover the chain algebra before Proposition 5.5. The reflection equation section is algebra-heavy: the constants C1 and C2 are said to be uniquely fixed, but the substitution that fixes them is not shown, and equations (9.17)-(9.18) contain a z/w inconsistency that looks like a typo. That section needs independent verification, not necessarily rejection.\n\nWho is this for: combinatorialists and integrable-model people working on Temperley-Lieb and Fuss-Catalan algebras. The paper deserves a serious referee. Send it to review with a request for a complete proof of Proposition 4.8, a careful inverse construction for Psi(r), and a cleaned-up Section 9. I would not cite it yet in my own work; I want the gap closed and the reflection solution checked first.","headline":"Useful new framework for Fuss-Catalan algebras via non-crossing partitions, but the central bijection rests on an unproved stacking step; worth refereeing, not desk-rejecting.","tokens_in":50070,"tokens_out":2983,"would_cite":false,"duration_ms":31924,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A15","06A07","16T25","81R12"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims the Fuss–Catalan algebra, the r-color Temperley–Lieb generalization, is realized on increasing r-chains of non-crossing partitions and is isomorphic to its chord-diagram action, yielding a new r=2 reflection solution.","keywords":["Fuss–Catalan algebras","Temperley–Lieb algebra","non-crossing partitions","generalized Dyck paths","chord diagrams","Kreweras endomorphism","increasing r-chains","reflection equation"],"falsifier":"Draw $\\Psi(1/2/3/4)$, $\\Psi(1/23/4)$, and $\\Psi(1234)$ by the rule of Section 3.3, place them on one set of $2\\cdot3\\cdot4$ points, and check whether the superposition is non-crossing and satisfies $i+j-1\\equiv 0 \\pmod{6}$; a crossing or a failed parity condition would refute Proposition 4.8 and break the isomorphism of Proposition 5.5. The same style of substitution settles the second claim: insert formulas (9.4)–(9.9) into the reflection equation (9.2) with explicit small diagrams and generic spectral parameters $w,z$.","tokens_in":48792,"feed_emoji":"🔗","tokens_out":21886,"duration_ms":190002,"temperature":0.7,"pith_summary":"The paper tries to establish that the Fuss–Catalan algebra — the $r$-color generalization of the Temperley–Lieb algebra, which acts on generalized ($r$-)Dyck paths — is realized just as directly on increasing $r$-chains of non-crossing partitions, one partition per color ordered by refinement. A bijection $\\Psi^{(r)}$ sends such chains to generalized chord diagrams, and the generators $F^{(s)}_i$, built from a block-merging operation conjugated by the Kreweras endomorphism, are shown to correspond exactly to the diagrammatic generators $E^{(s)}_i$, so the two algebras are isomorphic (Proposition 5.5). The same construction is extended to one- and two-boundary algebras acting on symmetric non-crossing partitions, with primed entries marking the symmetric chords that encode the second boundary, and these boundary algebras are also shown isomorphic to their diagrammatic versions. A new solution of the reflection equation for $r=2$ is derived from the boundary setting, providing concrete integrable boundary data. If the paper is right, the Fuss–Catalan algebra gains the same three-way description — paths, chord diagrams, and partitions — that underlies the Temperley–Lieb algebra's role in combinatorics and statistical mechanics.","feed_headline":"Fuss–Catalan algebras live on chains of non-crossing partitions","feed_subtitle":"The r-color Temperley–Lieb generalization matches its chord-diagram action and yields a new r=2 boundary solution.","key_machinery":"The central object is the bijection $\\Psi$ between non-crossing partitions of $[n]$ and chord diagrams on $2n$ points, defined by turning each block $(b_1,\\dots,b_p)$ into arches joining $b_i$ to $(b_{i+1}-1)'$ modulo $n$; under $\\Psi$ the Kreweras endomorphism $\\rho$ (the standard rotation on non-crossing partitions) becomes chord-diagram rotation, and the block-merging operators $F_i$ become the Temperley–Lieb generators $e_i$. For the $r$-color setting the machinery is the extended bijection $\\Psi^{(r)}$ from increasing $r$-chains to generalized chord diagrams on $2rn$ points obeying the parity condition $i+j-1\\equiv 0 \\pmod{2r}$, together with the generators $F^{(s)}_i = \\rho^{i-1} F^{(s)}_1 \\rho^{-(i-1)}$ built from the extended Kreweras endomorphism $\\rho(\\pi_1,\\dots,\\pi_r)=(\\rho(\\pi_r),\\dots,\\rho(\\pi_1))$. The superposition principle of Proposition 4.8 — chords coming from comparable partitions do not cross — is what lets one generalized chord diagram carry $r$ independent Temperley–Lieb actions, and it is what the isomorphism of Proposition 5.5 rests on; a second bijection $\\Phi$ described through cover-exclusive Dyck tilings is shown to differ from $\\Psi^{(r)}$ by a rotation.","core_discovery":"The central claim is Proposition 5.5: the Fuss–Catalan algebra $NC^{(r)}_n$ on increasing $r$-chains of non-crossing partitions, generated by operators $F^{(s)}_i$ formed from a block-merging map conjugated by the Kreweras endomorphism, is isomorphic to the diagrammatic Fuss–Catalan algebra $TL^{(r)}_n$ acting on generalized chord diagrams, by the correspondence $F^{(s)}_i \\mapsto E^{(s)}_i$. The fact underneath it is the superposition statement of Proposition 4.8: whenever $\\pi_1 \\leq \\pi_2 \\leq \\cdots \\leq \\pi_r$, the chord diagrams $\\Psi(\\pi_1),\\dots,\\Psi(\\pi_r)$ obtained entrywise from the bijection between non-crossing partitions and chord diagrams never intersect, so they stack into a single generalized chord diagram and the $r$ colors act independently. The paper also claims that the Kreweras endomorphism is conjugate, under the bijection, to rotation of chord diagrams (Proposition 3.11), that the cover relation of the non-crossing partition lattice records exactly when one chord diagram is obtained from another by a generator (Proposition 3.14), and that the one- and two-boundary Fuss–Catalan algebras act on symmetric non-crossing partitions, with primed integers marking the symmetric chords that encode the second boundary (Theorems 7.9, 7.10, 8.8, 8.12). Finally, Proposition 9.1 claims an explicit $r=2$ solution of the reflection equation, formulas (9.7)–(9.9), valid under the stated non-degeneracy conditions on $\\tau, \\tau_e, \\tau_o$, with the degenerate cases handled separately.","pith_inferences":["Editorial: if the isomorphism of Proposition 5.5 holds, known facts about the diagrammatic Fuss–Catalan algebra — its relations in Theorem 5.3, its dimensions, and its Yang–Baxter solution — transfer to the chain model; a check the paper leaves implicit is whether the chain representation is faithful for all $r$ and $n$.","Editorial: the construction uses only a lattice with a Kreweras-type rotation and comparable objects whose chord diagrams stay non-crossing, so the same recipe is likely to yield Fuss–Catalan-type algebras for other non-crossing families, such as non-crossing partitions of other Coxeter types.","Editorial: the $r=2$ reflection solution invites the same ansatz $K(w) = 1 + \\sum_s k_s(w) E^{(s)}_n$ for $r\\ge 3$; whether the functional equations admit solutions in the larger algebra is left open by the paper."],"forward_implications":["The Fuss–Catalan algebra acquires a purely combinatorial model — chains in the lattice of non-crossing partitions — so its elements and actions can be studied with partition combinatorics, in the same way the Temperley–Lieb algebra is studied with Dyck paths and chord diagrams.","Cover relations in the non-crossing partition lattice encode generator actions: $C_2 = e_i C_1$ holds exactly when the corresponding partitions cover one another in the appropriate direction, giving a poset-theoretic reading of the whole Temperley–Lieb action.","The dimension formulas $\\dim TL^{(r)}_n = |P^{(r)}_{n+1}|$, $\\dim 1\\text{-}BFC^{(r)}_n = B^{(r)}_{2n}$, and $\\dim 2\\text{-}BFC^{(r)}_n = K^{(r)}_n$ follow directly from the bijections, so counting chains, Dyck tilings, and folded diagrams yields the same numbers.","The boundary algebras act on symmetric non-crossing partitions, and the two-boundary case is carried by primed integers whose allowed positions obey the linear order (6.10) on symmetric chords — the second boundary is extra data on the same partitions.","The explicit $r=2$ solution (9.6)–(9.9) of the reflection equation furnishes concrete boundary weights, the ingredient needed to build an integrable lattice model with boundaries from the one-boundary Fuss–Catalan algebra."],"supporting_citations":[{"why":"Defines the Fuss–Catalan algebra as a generalization of the Temperley–Lieb algebra; this is the algebra whose chain model the paper constructs.","marker":"[5]"},{"why":"Gives the diagrammatic Fuss–Catalan algebra $TL^{(r)}_n$ and its Yang–Baxter solution, the representation that Proposition 5.5 matches and that the $r=2$ reflection solution builds on.","marker":"[14]"},{"why":"Establishes the bijection between increasing $r$-chains and trees that underlies counting chains by Fuss–Catalan numbers.","marker":"[11]"},{"why":"Provides the bijection between Dyck paths and non-crossing partitions that the chain bijection $\\kappa^{(r)}$ of Section 4.2 generalizes.","marker":"[34]"},{"why":"Introduces the Kreweras endomorphism, the rotation of non-crossing partitions used to move the generators around the circle.","marker":"[17]"},{"why":"Supplies the diagrammatic presentation of the Temperley–Lieb algebra whose generators $e_i$ match the operators $F_i$ under the bijection.","marker":"[16]"},{"why":"States the reflection equation whose $r=2$ solution is claimed in Proposition 9.1.","marker":"[31]"},{"why":"Introduces the two-boundary Temperley–Lieb algebra and the relation $(\\star1)$ that makes the two-boundary Fuss–Catalan algebra finite dimensional.","marker":"[8]"}],"fun_headline_variants":["Fuss-Catalan algebras act on generalized Dyck paths via non-crossing chains","Non-crossing partitions explain Fuss-Catalan algebras","Fuss-Catalan algebras solve reflection equation at r=2","Fuss-Catalan algebras generalize Temperley-Lieb via non-crossing chains","Fuss-Catalan algebras on Dyck paths gain non-crossing partition basis"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction rests on the claim that the chord diagrams belonging to the entries of an increasing chain always stack without crossings; the proof works out a single block-merging step in two configurations and extends to arbitrary chains by repeating the argument, so if any longer chain failed to stack, the central bijection and every isomorphism built on it would need repair.","fun_headline_variants_meta":{"raw":{"variants":["Fuss-Catalan algebras act on generalized Dyck paths via non-crossing chains","Non-crossing partitions explain Fuss-Catalan algebras","Fuss-Catalan algebras solve reflection equation at r=2","Fuss-Catalan algebras generalize Temperley-Lieb via non-crossing chains","Fuss-Catalan algebras on Dyck paths gain non-crossing partition basis"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001061,"raw_usage":{"total_tokens":4581,"prompt_tokens":1205,"completion_tokens":3376,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":821,"completion_tokens_details":{"reasoning_tokens":3277}},"tokens_in":821,"tokens_out":3376,"duration_ms":27671,"temperature":1.0,"reasoning_tokens":3277,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:44:59.604976+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Draw $\\Psi(1/2/3/4)$, $\\Psi(1/23/4)$, and $\\Psi(1234)$ by the rule of Section 3.3, place them on one set of $2\\cdot3\\cdot4$ points, and check whether the superposition is non-crossing and satisfies $i+j-1\\equiv 0 \\pmod{6}$; a crossing or a failed parity condition would refute Proposition 4.8 and break the isomorphism of Proposition 5.5. The same style of substitution settles the second claim: insert formulas (9.4)–(9.9) into the reflection equation (9.2) with explicit small diagrams and generic spectral parameters $w,z$.","supporting_citations":[{"cited_title":"Bisch and V","cited_arxiv_id":null,"evidence_quote":"Defines the Fuss–Catalan algebra as a generalization of the Temperley–Lieb algebra; this is the algebra whose chain model the paper constructs."},{"cited_title":"New Integrable Lattice Models From Fuss-Catalan Algebras","cited_arxiv_id":"hep-th/9807074","evidence_quote":"Gives the diagrammatic Fuss–Catalan algebra $TL^{(r)}_n$ and its Yang–Baxter solution, the representation that Proposition 5.5 matches and that the $r=2$ reflection solution builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the bijection between increasing $r$-chains and trees that underlies counting chains by Fuss–Catalan numbers."},{"cited_title":"More bijective Catalan combinatorics on permutations and on signed permutations","cited_arxiv_id":"0808.2822","evidence_quote":"Provides the bijection between Dyck paths and non-crossing partitions that the chain bijection $\\kappa^{(r)}$ of Section 4.2 generalizes."},{"cited_title":"Kreweras, Sur les partitions non croisees d’un cycle , Discrete Math","cited_arxiv_id":null,"evidence_quote":"Introduces the Kreweras endomorphism, the rotation of non-crossing partitions used to move the generators around the circle."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the diagrammatic presentation of the Temperley–Lieb algebra whose generators $e_i$ match the operators $F_i$ under the bijection."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the reflection equation whose $r=2$ solution is claimed in Proposition 9.1."},{"cited_title":"The two-boundary Temperley-Lieb algebra","cited_arxiv_id":"math/0703338","evidence_quote":"Introduces the two-boundary Temperley–Lieb algebra and the relation $(\\star1)$ that makes the two-boundary Fuss–Catalan algebra finite dimensional."}],"review_version":1}