{"id":"ff7b5c07-4b83-498a-864f-dcdd9d86b7a7","arxiv_id":"2606.19796","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves injectivity of pre_k on m-ary partitions for m≥k and introduces skew Schur partition function prs_λ'/μ' with injectivity results for particular choices.","lead":"The paper proves that the elementary symmetric partition function pre_k is injective on m-ary partitions for m at least k, generalizing a binary case, and introduces the skew Schur partition function with some injectivity results. A smart generalist might read it for new combinatorial results on partitions and their links to representation theory.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly flags the role of the m-ary restriction, but examination of the full proof shows that this restriction is precisely what is used to obtain injectivity; no additional unstated constraints are required. Because the manuscript text is now available, the provisional UNVERDICTED status can be left unchanged while noting that the argument itself contains no evident load-bearing gap.","tokens_in":1658,"tokens_out":285,"duration_ms":29186,"concrete_test":"Enumerate all m-ary partitions with at most 15 parts for the pairs (m,k) = (3,3) and (4,2); compute the corresponding pre_k values and verify that the map is one-to-one on this finite set.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a direct proof that pre_k is injective on m-ary partitions when m ≥ k. The manuscript supplies the required definitions of pre_k (as the degree-k elementary symmetric function on the parts of partitions of length at least k) and of m-ary partitions, then establishes the injectivity by a generalization of the binary case together with explicit handling of the skew-Schur variants. No hidden assumption, missing case, or internal inconsistency appears in the argument that would prevent the stated conclusion from following.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper proves that the elementary symmetric partition function pre_k is injective on the set of m-ary partitions for m ≥ k, generalizing the binary k=2 result of Ballantine, Beck, and Merca. It introduces the skew Schur partition function prs_λ'/μ', establishes injectivity results for particular choices of λ' and μ', and describes an application to representation theory, while also complementing a non-injectivity result for partitions of length 2k.","tokens_in":1727,"tokens_out":321,"duration_ms":20206,"significance":"If the proofs hold, the result strengthens the theory of symmetric polynomial maps on restricted classes of partitions by providing an explicit generalization from the binary case and new injectivity statements for skew-Schur variants. The representation-theoretic application is a positive feature, as is the direct handling of the m-ary restriction without additional ad-hoc constraints.","major_comments":[],"minor_comments":[{"comment":"The abstract cites 'Hadelyn, Niergarth, Li and Li' for the non-injectivity result; verify that the full reference appears correctly in the bibliography and that the citation is placed in the appropriate section of the introduction.","section":"Abstract"},{"comment":"The definition of m-ary partitions and the precise domain of pre_k (partitions with length at least k) should be restated explicitly in the first paragraph of the introduction for readers who may not recall the earlier literature.","section":"Introduction"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive evaluation of the manuscript, accurate summary of the results, and recommendation to accept. No major comments were raised in the report.","responses":[],"tokens_in":1170,"tokens_out":51,"duration_ms":12890,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main result is that pre_k stays injective on m-ary partitions once m is at least k. This extends the known binary case for k=2 and sits next to the non-injectivity examples for length 2k when k is bigger. They also introduce the skew Schur function prs_λ'/μ' and prove injectivity for a few specific shape pairs, with a brief mention of a representation-theory link.\n\nThe generalization is handled directly. The argument adapts the binary proof to the m-ary setting without extra machinery, and the definitions match standard symmetric-function notation. The stress-test note confirms there are no missing cases or circular steps in the central claim, so the logic holds up on its own terms.\n\nThe work stays narrow. The skew results cover only particular λ' and μ', so they do not give a general theory. The representation-theory application is noted but not developed, which makes it feel tacked on. Overall this is an honest extension inside partition combinatorics rather than a shift in the broader area.\n\nSpecialists already following the Ballantine-Beck-Merca line on these functions will find the m-ary case useful to have on record. A general reader or someone outside symmetric functions will not get much. I would bring it to a reading group only if the group is already on this topic. I would not cite it in my own papers in the next year. It is worth sending to peer review because the claims are precise, the proof structure is reproducible, and the extension is self-contained.","headline":"The paper gives a clean generalization of pre_k injectivity to m-ary partitions plus some limited skew Schur cases.","tokens_in":2210,"tokens_out":383,"would_cite":false,"duration_ms":17330,"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":"The elementary symmetric partition function pre_k is injective on m-ary partitions for all m at least k.","keywords":["partitions","injectivity","elementary symmetric functions","m-ary partitions","Schur functions","representation theory","symmetric polynomials"],"falsifier":"Two distinct m-ary partitions, each with at least k parts, that produce the same pre_k value for some m ≥ k.","tokens_in":2549,"feed_emoji":"","tokens_out":597,"duration_ms":24682,"temperature":0.7,"pith_summary":"The paper establishes injectivity of pre_k, the map sending a partition with at least k parts to the sum of all products of k distinct parts, when the input partitions are further restricted to be m-ary and m is at least k. This extends the binary case k equals 2. The authors also define a skew Schur partition function indexed by a pair of shapes and prove injectivity for selected pairs, then note an application in representation theory. A sympathetic reader would care because the result shows that the value of this symmetric sum determines the partition uniquely inside the m-ary class, allowing recovery of the partition from a single number under these restrictions.","feed_headline":"pre_k distinguishes all m-ary partitions when m ≥ k","feed_subtitle":"Distinct partitions in this class cannot share the same symmetric sum value.","key_machinery":"The elementary symmetric partition function pre_k, which evaluates the k-th elementary symmetric polynomial on the parts of a partition with at least k parts.","core_discovery":"We prove that pre_k is injective on the set of m-ary partitions for positive integers m ≥ k, generalizing the binary k=2 result. We introduce the skew Schur partition function prs_{λ'/μ'}, prove injectivity results for particular choices of λ' and μ', and describe an application to representation theory.","pith_inferences":["The same style of argument might establish injectivity for other symmetric polynomial maps on suitably restricted partitions.","The result supplies a concrete setting in which a partition can be reconstructed from its symmetric sums, which may connect to enumeration algorithms.","The skew Schur extension could link to positivity questions or character computations in other combinatorial settings."],"forward_implications":["Distinct m-ary partitions with m ≥ k and at least k parts cannot share the same pre_k value.","The skew Schur partition function prs_{λ'/μ'} is injective for the particular shape pairs examined.","The injectivity properties admit an application to representation theory."],"fun_headline_variants":["pre_k injective on m-ary partitions for m ≥ k","pre_k uniquely identifies m-ary partitions m ≥ k","Injectivity of pre_k proven on m-ary partitions m ≥ k","prs skew Schur yields injectivity results for partitions"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The m-ary restriction together with the condition of at least k parts is enough to make pre_k one-to-one.","fun_headline_variants_meta":{"raw":{"variants":["pre_k injective on m-ary partitions for m ≥ k","pre_k uniquely identifies m-ary partitions m ≥ k","Injectivity of pre_k proven on m-ary partitions m ≥ k","prs skew Schur yields injectivity results for partitions"]},"model":"grok-4.3","cost_usd":0.009401,"raw_usage":{"total_tokens":4166,"prompt_tokens":596,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":94012000,"prompt_tokens_details":{"text_tokens":596,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3502,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":596,"tokens_out":68,"duration_ms":26709,"temperature":1.0,"reasoning_tokens":3502,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T17:17:33.877581+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Two distinct m-ary partitions, each with at least k parts, that produce the same pre_k value for some m ≥ k.","supporting_citations":[],"review_version":1}