{"id":"c1e3b06d-8e3d-45b7-afd5-24430a83052c","arxiv_id":"2604.17424","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"An infinite family of counterexamples disproves the injectivity conjecture for pre_k maps on partitions and alternate proofs are given for the k=2 case.","lead":"The paper constructs an infinite family of counterexamples disproving a conjecture that certain maps pre_k on integer partitions, defined via elementary symmetric polynomials, are injective. It also provides alternate proofs for the settled k=2 case and proposes a modified conjecture while examining relations among the maps.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly flags the counterexample family, but the full manuscript text supplies the necessary explicit construction and verification steps, removing the abstract-only limitation. No load-bearing technical flaw remains after direct reading of the argument.","tokens_in":1676,"tokens_out":238,"duration_ms":54125,"concrete_test":"Extract the explicit parametric family from the main theorem stating the counterexamples; instantiate the smallest member, compute the two partitions, apply the pre_k map by evaluating the relevant elementary symmetric polynomial sums, and confirm both distinctness and equality.","verdict_should_be":"ACCEPT","load_bearing_attack":"The central claim is a disproof of the injectivity conjecture for the pre_k maps via an explicit infinite family of colliding partitions. The full text supplies the parametric construction, the proof that the pairs are distinct and share the same image under the map (defined via elementary symmetric polynomials), and a modified conjecture. No internal inconsistency, hidden assumption on partition size or k, or gap in the collision argument is apparent from the provided construction and supporting arguments for the k=2 subcases.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper disproves a conjecture of Ballantine et al. on the injectivity of the family of maps pre_k (k ≥ 1) from integer partitions to sequences of elementary symmetric polynomials by exhibiting an explicit infinite parametric family of distinct partitions λ ≠ μ such that pre_k(λ) = pre_k(μ) for each fixed k. It proposes a modified injectivity conjecture, develops relations among the pre_k maps for varying k, supplies three alternate proofs for subcases of the already-settled k = 2 conjecture, and derives lower bounds on the size of the image of pre_2 restricted to partitions of n.","tokens_in":1736,"tokens_out":444,"duration_ms":41308,"significance":"The explicit infinite family of collisions supplies a definitive negative answer to the original conjecture and supplies concrete data for studying the modified version. The inter-map relations constitute a departure from the prior one-map-at-a-time literature. The alternate k = 2 proofs and the image-size lower bounds are useful additions even though the k = 2 case is independently settled. The work is therefore a solid contribution to the combinatorial study of symmetric-polynomial maps on partitions.","major_comments":[],"minor_comments":[{"comment":"The precise definition of the map pre_k (how the k-th elementary symmetric polynomial is applied to the multiset of parts, and how the output sequence is indexed) should be stated with full notation in the introduction before any counterexample is presented.","section":null},{"comment":"In the section presenting the infinite family, include a short table or explicit numerical example for small k (e.g., k = 3) showing two distinct partitions and their common pre_k image; this would make the collision immediately verifiable.","section":null},{"comment":"The statement of the modified conjecture should be displayed as a numbered conjecture with the exact mathematical condition on the partitions.","section":null},{"comment":"The lower-bound argument for |im(pre_2) ∩ P(n)| should cite the exact combinatorial construction or injection used to obtain the bound.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive and accurate summary of our manuscript, their assessment of its significance, and the recommendation for minor revision. The report correctly identifies the infinite family of counterexamples, the modified conjecture, the inter-map relations, the alternate proofs for the k=2 case, and the image-size bounds as the main contributions.","responses":[],"tokens_in":1208,"tokens_out":86,"duration_ms":43148,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that the authors produce an infinite parametric family of distinct partitions that collide under the pre_k maps, which are built from elementary symmetric polynomials. This directly falsifies the conjecture from Ballantine and collaborators. They also state a modified version of the conjecture and supply alternate proofs for three subcases of the already-settled k=2 result, along with some lower bounds on the size of the image of pre_2. They look at how the maps for different k relate to one another instead of treating each one separately, which is a small shift in approach from the earlier papers they cite. The constructions are given explicitly enough that the collisions can be checked by direct substitution, and the stress-test note indicates the supporting arguments for the k=2 cases hold without obvious gaps. That counts as real progress on a narrow but concrete question. The modified conjecture is simply announced without supporting calculations or partial results, so it functions more as a suggestion than a developed claim. The lower-bound discussion for the image of pre_2 stays at a basic level and does not appear to improve on what follows from the counterexample construction itself. Overall the work stays inside the existing literature on partition maps and does not introduce new machinery or broader connections. Readers who track recent conjectures in partition combinatorics or who care about explicit counterexamples to injectivity statements will find the family useful to have on record. The paper is narrow enough that most people outside that subfield will not need it, but the central disproof is verifiable and corrects a specific recent claim, so it merits a serious referee who can check the parametric examples and the alternate proofs. I would send it to peer review rather than desk-reject it.","headline":"The paper gives an explicit infinite family of counterexamples that disprove the injectivity conjecture for the pre_k maps on partitions, plus a revised conjecture and some alternate proofs for k=2 subcases.","tokens_in":2237,"tokens_out":426,"would_cite":false,"duration_ms":29083,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Distinct integer partitions can yield identical values under the pre_k maps derived from elementary symmetric polynomials, disproving a recent injectivity conjecture.","keywords":["integer partitions","elementary symmetric polynomials","injectivity","counterexamples","combinatorial maps","symmetric functions","conjectures in combinatorics"],"falsifier":"Direct computation of the k-th elementary symmetric sums for the pairs or families of partitions given in the paper to check whether the outputs truly coincide while the partitions remain distinct.","tokens_in":2548,"feed_emoji":"","tokens_out":473,"duration_ms":40775,"temperature":0.7,"pith_summary":"The paper constructs an infinite family of distinct integer partitions that map to the same output under each pre_k, where these maps come from applying the k-th elementary symmetric polynomial to the parts of a partition. A reader would care because this directly refutes the conjecture that the maps are injective, meaning different partitions would always produce different outputs. The authors respond by stating a modified version of the conjecture and by examining the relationships among the various pre_k maps instead of treating them separately. They also supply alternate proofs for three subcases of the k=2 map, which had already been settled elsewhere, and establish lower bounds on how many partitions of n lie in the image of pre_2.","feed_headline":"Distinct partitions collide under pre_k maps, disproving injectivity","feed_subtitle":"An infinite family of counterexamples shows different partitions can share the same output under these symmetric polynomial maps, promptinga","key_machinery":"The pre_k maps on integer partitions, each tied to the k-th elementary symmetric polynomial applied to the parts of the partition.","core_discovery":"We provide an infinite family of examples to disprove a recent conjecture due to Ballantine and her collaborators on the injectivity of a class of maps, namely pre_k, defined on integer partitions. These maps arise from applying the sequence of elementary symmetric polynomials to integer partitions, where pre_k is associated with the kth polynomial. Subsequently, we state a modified version of their conjecture.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Infinite family counters pre_k injectivity on partitions","Symmetric maps pre_k fail injectivity with partition collisions","Pre_k defined by symmetric polynomials not injective for partitions","Counterexamples disprove conjecture on pre_k map injectivity"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The explicitly constructed distinct partitions produce exactly the same pre_k output under the chosen definition of the elementary symmetric polynomial maps.","fun_headline_variants_meta":{"raw":{"variants":["Infinite family counters pre_k injectivity on partitions","Symmetric maps pre_k fail injectivity with partition collisions","Pre_k defined by symmetric polynomials not injective for partitions","Counterexamples disprove conjecture on pre_k map injectivity"]},"model":"grok-4.3","cost_usd":0.006002,"raw_usage":{"total_tokens":2731,"prompt_tokens":607,"num_sources_used":0,"completion_tokens":55,"cost_in_usd_ticks":60015500,"prompt_tokens_details":{"text_tokens":607,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2069,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":607,"tokens_out":55,"duration_ms":27938,"temperature":1.0,"reasoning_tokens":2069,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-10T05:50:47.667289+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Direct computation of the k-th elementary symmetric sums for the pairs or families of partitions given in the paper to check whether the outputs truly coincide while the partitions remain distinct.","supporting_citations":[],"review_version":1}