{"id":"0e57f35f-434c-4d5d-b023-a7e81574171c","arxiv_id":"2502.09161","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Switching left and right branches in binary trees yields new bijections on weakly increasing trees, reproving and generalizing known symmetries on plane trees and 312-avoiding permutations.","lead":"This paper builds four bijections on weakly increasing trees by switching left and right branches in corresponding binary trees. It provides bijective proofs of known plane-tree symmetries, a non-recursive version of Deutsch's bijection, and new symmetries on 312-avoiding permutations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The statistic translations in Lemmas 2.2 and 2.4 are load-bearing and are delegated to the reader; a single miscount would collapse Theorems 1.2 and 1.7. A finite brute-force check of small multisets should settle them.","rationale":"The reader's weakest_assumption is exactly the statistic translation in Lemmas 2.2 and 2.4: those lemmas are load-bearing for Theorems 1.2, 1.4, and 1.7, and their proofs are sketched or omitted. I independently examined the surrounding argument and found no more likely point of failure. In particular, the proofs of Theorem 1.7 and Theorem 2.11 are comparatively detailed once the lemma translations are granted, and the Riccati equation in Section 3 appears internally consistent after checking boundary effects. The one genuinely exposed part of the paper is the correspondence ρ and the six/refined-six statistic identifications. The concern is not that the claims are false—they may well be correct—but that the paper's own text defers the verifying computations, and the formulation of ρ with 'only if' conditions is ambiguous enough that an implementer cannot immediately reconstruct the map. The proposed concrete test is a direct, finite verification that settles the concern without relying on the authors' assertion. Since the reader's verdict is already CONDITIONAL with the same weakest assumption, my pass does not change that verdict; it strengthens the reason for the condition by identifying a concrete check that should be supplied or performed.","tokens_in":18244,"tokens_out":20732,"duration_ms":210610,"concrete_test":"Enumerate all weakly increasing trees for small multisets, starting with M ∈ {{1^2}, {1^3}, {1,2}, {1^2,2}, {1,2,2}, {1^3,2}}, and implement ρ exactly from its definition. For each T, compute sleaf, eleaf, yleaf, yint and the six refined statistics on T, and compare them with the asserted binary-tree configurations in Lemmas 2.2 and 2.4. This is a finite check: also apply Φ = ρ^{-1} ∘ φ ∘ ρ and Ψ = ρ^{-1} ∘ ψ ∘ ρ to every tree in these sets and verify the claimed statistic exchanges. If every equality holds through |M| = 8, the central construction is substantially validated; if any equality fails, the corresponding theorem is false as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Every main bijection in Section 2 routes through the correspondence ρ and the statistic translations in Lemmas 2.2 and 2.4. Theorem 1.2 uses Lemma 2.2 verbatim: φ switches both children at every node, and the claimed exchange (sleaf, eleaf, yleaf, yint) ↔ (eleaf, sleaf, yint, yleaf) is only as sound as those four identities. Theorem 1.7 similarly rests on the six refined identities in Lemma 2.4, and the later permutation applications inherit them through Lemma 3.1. Yet Lemma 2.2 is proved by a one-line sketch for (iv) and 'follow easily' for the rest, and Lemma 2.4 says only that all assertions 'can be verified routinely' and leaves details to the reader. This is not a cosmetic issue: the paper's definition of ρ is given by two 'only if' conditions, which are ambiguous as written and do not explicitly prescribe the full edge set of ρ(T). In particular, Lemma 2.2(iv) asserts that young internal nodes correspond to nodes with only a right child, justified by an 'internal eldest node' phrase that is itself hard to parse. If any one of these local correspondences is off by a single configuration, then φ or ψ would no longer induce the stated statistic transformation, and the central theorems would fail. Because the lemmas are stated without proof, the internal consistency of the construction is currently unverified, not merely unpolished.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies weakly increasing trees on a multiset, a common generalization of plane trees and increasing trees. It defines a correspondence ρ to weakly increasing binary trees and then constructs involutions Φ, Ψ and a map Θ by switching left and right branches of selected nodes in binary trees. The main claims are: Φ exchanges (sleaf, eleaf, yleaf, yint) with (eleaf, sleaf, yint, yleaf) (Theorem 1.2); Ψ exchanges (suleaf, entleaf) while preserving the other four refined leaf statistics (Theorem 1.7); Θ is a non-recursive realization of Deutsch's bijection (Theorem 2.13); and via the binary-tree encoding of permutations, the maps Λ and Υ give symmetries in 312-avoiding permutations, including a generalization of Kitaev-Zhang's result (Theorems 3.6 and 3.7). A group-action proof of a generating function identity (Theorem 1.4) and a new algebraic generating function for refined Narayana polynomials (Theorem 2.11) are also included.","tokens_in":18508,"tokens_out":15083,"duration_ms":150321,"significance":"If the constructions are made fully precise, the paper would provide the requested bijective proofs of the symmetries of Dong et al. in a unified weakly increasing tree setting, a non-recursive version of Deutsch's bijection, and new equidistribution results for plane trees and 312-avoiding permutations. The explicit nature of the maps and the concrete generating-function calculations are strengths, and the claimed results are concrete enough to be checked exhaustively on small multisets. The main obstacle is not the plausibility of the statements but the under-specification of the central correspondence ρ and the delegation of key statistic translations to the reader.","major_comments":[{"comment":"The map ρ is not well-defined as written. Clauses (i) and (ii) are phrased as necessary conditions ('only if'), so they do not determine a unique image. If they are intended as equivalences, the conditions still omit the leftmost child of every parent that has at least two children: such a child is neither a rightmost child nor a node with a closest elder sibling, so it would have no incoming edge in ρ(T). Since all later assertions, starting with Lemma 2.2, are statements about ρ(T), the definition must be replaced by an explicit recursive rule, such as the standard first-child/next-sibling correspondence or its mirror image, and the statistic translations must be verified against that rule.","section":"Section 2.1, Definition 2.1"},{"comment":"The four identities in Lemma 2.2 are load-bearing: Theorem 1.2 and the translation in equation (2.1) both use them verbatim. However, only part (iv) receives an argument, and that argument refers to an undefined 'internal eldest node'; parts (i)-(iii) are asserted to 'follow easily'. Please supply a complete proof, for example by displaying the local binary-tree configuration corresponding to each child configuration of a node in T, and verify each of the four statistics explicitly.","section":"Lemma 2.2"},{"comment":"All six refined statistic identities in Lemma 2.4 are delegated to the reader with the statement that they 'can be verified routinely'. These identities are the exact mechanism by which Ψ is shown in Theorem 1.7 to have its claimed statistic effect, and they are also used in Section 2.4 to identify the polynomials Nn with sums over binary trees. This is a central translation step, not a cosmetic omission; each of the six identities should be proved explicitly from the corrected definition of ρ.","section":"Lemma 2.4"},{"comment":"The proof of Θ(T)=p(T) is a single sentence saying that the induction is straightforward and left to the interested reader. Since equality with the known recursive Deutsch map is one of the main deliverables of Section 2.5, the induction should be written out in full. Lemma 2.14 gives a description of the inverse of θ on binary trees, but it does not by itself establish equality of Θ and p on weakly increasing trees.","section":"Theorem 2.13"}],"minor_comments":[{"comment":"The displayed conclusion 'An,2 = (n−1)z2An−2' should read 'An,n−2 = (n−1)z2An−2'. In addition, case (5) should be restricted to 3 ≤ i ≤ n−3 so that it does not overlap with the separately treated case i = n−2.","section":"Theorem 3.2, case (4)"},{"comment":"The phrase 'c2 is an arbitrary positive constant' should be 'arbitrary constant'. The tracking of the constants c1 and c2 before deriving formula (3.3) is also too terse; please show explicitly how the initial conditions A(z;0)=0 and A1(z;0)=z determine them.","section":"Proof of Proposition 3.3"},{"comment":"There are several typographical errors, including 'Motived' in the abstract, 'weak ly' in the abstract, and 'tripe' in the statement of Theorem 3.6. These should be corrected in a revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the results are plausible and likely correct, but the formal definition of ρ is genuinely incomplete as written, and the statistic translations in Lemmas 2.2 and 2.4 are delegated rather than proved. These are load-bearing for the main theorems, but they are fixable within the scope of a revision, so I recommend major revision rather than rejection. I would also ask that the authors state clearly which plane-tree-to-binary-tree convention is used, since the current 'only if' language makes it impossible for a reader to test even a small example of Lemma 2.2."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper earns its place. It unifies three known symmetry results of Dong et al. under one construction, gives bijective proofs where only algebraic proofs existed, and adds new generating functions plus a non-recursive Deutsch bijection. The binary-tree switching operation is a legitimate unifying device, not a repackaged trick. Theorems 1.2, 1.7, 2.13, 3.6, and 3.7 are real extensions, and the group-action proof of Theorem 1.4 is clean. The route from tree statistics to permutation patterns via Lemma 3.1 is standard and sound.\n\nThe main caveat is exactly what the stress-test note says: Lemmas 2.2 and 2.4 carry all of Section 2, and their proofs are sketched. Lemma 2.2(iv) gets one sentence; Lemma 2.4 says \"verified routinely\" and leaves the details out. If any one of the six local correspondences is miscounted, Theorems 1.2 and 1.7 collapse. I do not think they are miscounted—the worked examples and the later generating function in Theorem 2.11 are consistent—but the authors should expand the proofs, especially for Lemma 2.4. This is a request for more detail, not a sign the theorems are wrong. The same goes for Theorem 2.13: the induction is plausible and Lemma 2.14 gives the key step, but \"left to the interested reader\" is too thin for a theorem advertised as making Deutsch's bijection non-recursive.\n\nMinor issue: in Theorem 3.2, case (4) writes An,2 where it should be An,n-2. The surrounding text makes the intent clear, so this is a typo, not a mathematical problem.\n\nOn reproducibility: there is no code or data, but the constructions are explicit enough that a small brute-force check over n up to 6 or 7 would settle the lemmas. I would encourage the authors to include such a check or a fully expanded proof. The citation pattern is appropriate; the earlier work by Lin, Ma, Ma, and Zhou is credited correctly.\n\nWho this is for: enumerative combinatorics researchers working on tree bijections, refined Narayana and Motzkin polynomials, and permutation patterns. It deserves a serious referee. If I were editing, I would send it out: the mathematics is very likely correct, and the requested revisions are about completeness, not about fixing a broken argument.","headline":"A solid, genuinely useful bijective-combinatorics paper, with the main caveat that some load-bearing lemmas are left as sketches.","tokens_in":19116,"tokens_out":2238,"would_cite":true,"duration_ms":23809,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A05","05A15","05C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Branch-swap maps prove four tree bijections.","keywords":["weakly increasing trees","binary trees","bijections","involutions","plane trees","permutation patterns","Narayana polynomials","Motzkin polynomials"],"falsifier":"Enumerate all weakly increasing trees on a small multiset such as {1,1,2,2}, compute for each tree the statistics sleaf, eleaf, yleaf, and yint, and compare them with the numbers of right leaves, left leaves, single-left-child nodes, and single-right-child nodes in the binary tree ρ(T). A single tree where the number of singleton leaves is not the number of right leaves in ρ(T) would settle Lemmas 2.2 and, with them, Theorem 1.2.","tokens_in":18000,"feed_emoji":"🌳","tokens_out":4338,"duration_ms":42512,"temperature":0.7,"pith_summary":"This paper builds four bijections on weakly increasing trees, the common generalization of plane trees and increasing trees, using one simple operation: translate a tree to a binary tree, switch left and right subtrees at chosen nodes, and translate back. The main theorems give exact transformations of refined statistics, yielding bijective proofs of plane-tree symmetries that were previously established only through generating functions. A direct, non-recursive construction of Deutsch's plane-tree bijection follows. The same switch operation transfers to 312-avoiding permutations, producing new involutions that generalize a symmetry of Kitaev and Zhang.","feed_headline":"One branch switch proves four tree bijections","feed_subtitle":"A single binary-tree operation yields bijective proofs of symmetries on weakly increasing trees.","key_machinery":"The central object is the natural correspondence ρ between weakly increasing trees and weakly increasing binary trees, together with the node-wise operation φ_x that switches the left and right branches of a chosen node x. Because the switches at different nodes commute, one can switch all nodes in a prescribed set at once. The statistic proofs reduce to lemmas stating that tree statistics such as singleton leaves, elder leaves, young leaves, and young internal nodes become right leaves, left leaves, and single-child nodes in the binary tree.","core_discovery":"The central claim is that the involutions Φ, Ψ, and Θ, defined as conjugates of branch-switching maps on binary trees, have precisely controlled effects on six refined leaf and internal-node statistics. Φ sends (sleaf, eleaf, yleaf, yint) to (eleaf, sleaf, yint, yleaf); Ψ preserves (snuleaf, etleaf, syleaf, yerleaf) and exchanges (suleaf, entleaf); and Θ equals Deutsch's recursive bijection on plane trees, now described non-recursively. Permutation analogues Λ and Υ act on 312-avoiding permutations, with Υ exchanging the consecutive patterns 1324 and 3241 while preserving peaks, double descents, and double ascents. These results provide bijective proofs of the Dong–Du–Ji–Zhang symmetries and extend them to all weakly increasing trees.","pith_inferences":["The branch-switching recipe is likely more general: any statistic expressible as a local binary-tree configuration should be swappable by an analogous involution, suggesting many further equidistributions beyond the four stated.","Because λ maps 312-avoiding permutations to binary trees with labels in preorder, the same switch maps probably realize additional consecutive pattern symmetries; one could search for length-5 patterns paired by Υ or Λ.","The non-recursive form of Deutsch's bijection via heads gives a level-by-level interpretation of the map that may extend to other Catalan-object families beyond plane trees."],"forward_implications":["The symmetries of Dong, Du, Ji and Zhang on plane trees, originally proved algebraically, now have bijective proofs that extend to every multiset of labels.","Deutsch's recursive plane-tree bijection is realized in one non-recursive step, namely switching branches at all heads (root nodes and right children) of the corresponding binary tree.","The pair (oleaf, elint) is symmetric on plane trees, and the number of plane trees with elint = k is given by the explicit formula in Corollary 2.16.","The involution Λ on 312-avoiding permutations gives an involution proof of the Kitaev–Zhang symmetry, while Υ exchanges the consecutive patterns 1324 and 3241.","Theorem 1.4 and its consequence give a group-action proof that the number of weakly increasing trees with oleaf = k and even yleaf equals the number with odd yleaf."],"supporting_citations":[{"why":"Introduces weakly increasing trees and gives their cardinality formula; the object class on which all bijections are defined.","marker":"[19]"},{"why":"Proves the plane-tree symmetries that the paper aims to establish bijectively and asks for bijective proofs.","marker":"[11]"},{"why":"Provides the recursive extension of Deutsch's bijection to weakly increasing trees, which Θ reproduces non-recursively.","marker":"[22]"},{"why":"Defines Deutsch's original recursive plane-tree bijection, which Theorem 2.13 realizes in one non-recursive step.","marker":"[9]"},{"why":"States the Kitaev–Zhang symmetry on 312-avoiding permutations that Λ generalizes.","marker":"[16]"},{"why":"Introduces the old-leaf and young-leaf statistics that underlie the refined six statistics used throughout.","marker":"[4]"},{"why":"Supplies the mapping λ between permutations and increasing binary trees used to transfer the tree involutions to permutation statistics.","marker":"[25]"},{"why":"Provides the classical plane-tree-to-binary-tree correspondence of which ρ is the extension.","marker":"[26]"}],"fun_headline_variants":["One branch switch, four tree bijections","Flip one branch, get four bijections","Four tree bijections from a single flip","Branch swap unlocks tree symmetries","Single swap yields four bijections"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's load-bearing premise is that the refined leaf and internal-node statistics on a weakly increasing tree are counted exactly by the corresponding local configurations in its associated binary tree, as stated in Lemmas 2.2 and 2.4; if any of these translations is off by one, the involutions no longer have the claimed effect.","fun_headline_variants_meta":{"raw":{"variants":["One branch switch, four tree bijections","Flip one branch, get four bijections","Four tree bijections from a single flip","Branch swap unlocks tree symmetries","Single swap yields four bijections"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000378,"raw_usage":{"total_tokens":1949,"prompt_tokens":824,"completion_tokens":1125,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":440,"completion_tokens_details":{"reasoning_tokens":1059}},"tokens_in":440,"tokens_out":1125,"duration_ms":10884,"temperature":1.0,"reasoning_tokens":1059,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T22:26:11.884277+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all weakly increasing trees on a small multiset such as {1,1,2,2}, compute for each tree the statistics sleaf, eleaf, yleaf, and yint, and compare them with the numbers of right leaves, left leaves, single-left-child nodes, and single-right-child nodes in the binary tree ρ(T). A single tree where the number of singleton leaves is not the number of right leaves in ρ(T) would settle Lemmas 2.2 and, with them, Theorem 1.2.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces weakly increasing trees and gives their cardinality formula; the object class on which all bijections are defined."},{"cited_title":"Dong, L.R","cited_arxiv_id":null,"evidence_quote":"Proves the plane-tree symmetries that the paper aims to establish bijectively and asks for bijective proofs."},{"cited_title":"Lin and J","cited_arxiv_id":null,"evidence_quote":"Provides the recursive extension of Deutsch's bijection to weakly increasing trees, which Θ reproduces non-recursively."},{"cited_title":"Deutsch, A bijection on ordered trees and its conseque nces, J","cited_arxiv_id":null,"evidence_quote":"Defines Deutsch's original recursive plane-tree bijection, which Theorem 2.13 realizes in one non-recursive step."},{"cited_title":"Kitaev and P.B","cited_arxiv_id":null,"evidence_quote":"States the Kitaev–Zhang symmetry on 312-avoiding permutations that Λ generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the old-leaf and young-leaf statistics that underlie the refined six statistics used throughout."},{"cited_title":"Stanley, Enumerative Combinatorics , vol","cited_arxiv_id":null,"evidence_quote":"Supplies the mapping λ between permutations and increasing binary trees used to transfer the tree involutions to permutation statistics."},{"cited_title":"Stanley, Catalan numbers , Cambridge University Press, New York, 2015","cited_arxiv_id":null,"evidence_quote":"Provides the classical plane-tree-to-binary-tree correspondence of which ρ is the extension."}],"review_version":1}