{"id":"816b9c08-5068-4ee5-b31b-b27dac6094e8","arxiv_id":"2607.29369","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Automated transfer-matrix enumeration yields rational generating functions for L-tetromino tilings of k×n boards for k=5,...,9, with several sequences not currently in the OEIS.","lead":"This paper uses a known automated transfer-matrix method to count tilings of 5- to 9-wide boards by L-shaped tetrominoes, producing rational generating functions and new integer sequences. It first confirms the method by recovering a known generating function for 4-wide boards.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Displayed recurrences have off-by-one errors (Prop. 2 gives a_4=88 vs. the GF's 86), so the new GFs/sequences rest on program output with demonstrably unchecked presentation.","rationale":"The reader's weakest assumption was the correctness of the Maple/transfer-matrix implementation, which is indeed the core risk. My review supplies a concrete, verifiable manifestation of that risk: the paper's displayed recurrences have systematic off-by-one errors. This strengthens the need for independent verification but does not conclusively falsify the generating functions, since the GFs themselves would still be correct if only the recurrence index bounds were off. The missing k=9 GF further weakens the paper's support for its largest new sequence. Because the appropriate verdict is already CONDITIONAL, my finding does not change the verdict; it refines the condition: the authors should fix the recurrences, supply the missing k=9 formula, and provide reproducible code plus an independent cross-check.","tokens_in":8299,"tokens_out":9652,"duration_ms":88136,"concrete_test":"Expand each displayed GF (Props. 2, 3, 4, 6, 7, 8) to at least 30 terms and compare with the stated first terms and recurrences; the off-by-one error in Prop. 2 is immediately visible at n=4. Then write an independent brute-force backtracking or transfer-matrix program for L-tetromino tilings of 5×8n, 6×4n, 7×8n, 8×n, 9×8n for n≤4 and check every table entry. If any coefficient differs, the central exactness claim fails. Also request the missing k=9 generating function for 9×8n.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that Zeilberger's program produces exact rational generating functions and new sequences for L-tetromino tilings of widths 5–9. The new results are unverified program outputs, and the paper's own \"Consequently\" recurrences are demonstrably wrong at the stated index bounds.\n\nProp. 2 gives GF (2t^4+2t^3+t^2−1)/(4t^4+4t^3+4t^2+2t−1), equivalently (1−t^2−2t^3−2t^4)/(1−2t−4t^2−4t^3−4t^4). Expanding gives a_0=1, a_1=2, a_2=7, a_3=24, a_4=86. But the stated recurrence a_n=2a_{n−1}+4a_{n−2}+4a_{n−3}+4a_{n−4}, n≥4, with the listed initial values gives 2·24+4·7+4·2+4·1 = 88, contradicting the GF and the paper's own first terms. The correct bound is n≥5, not n≥4.\n\nThe same off-by-one error appears in Prop. 3 (8×4n, T-tetromino): stated recurrence with n≥2 gives a_2=69, while the GF gives 84; the correct bound is n≥3. In Prop. 4 (12×4n, T-tetromino), n≥4 gives a_4=5,249,205 instead of 5,253,822; the correct bound is n≥5. Thus every displayed recurrence in the paper has the same boundary error.\n\nThis does not by itself prove the generating functions are wrong, but it shows the manuscript has not been checked against its own formulas, so the unverified GFs and sequences for k=5–9 cannot be taken at face value. Moreover, the promised k=9 generating function for 9×8n is absent from Appendix A.2, leaving the largest new sequence with no formula to verify.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper revisits Zeilberger's Maple-based transfer-matrix method for automatic enumeration of polyomino tilings and applies it to L-tetromino tilings of k×n boards. It first recovers the known generating function for 4×2n boards (Bělohoubek–Slavík), then reports rational generating functions and integer sequences for widths k=5,...,9, several claimed new to the OEIS. T-tetromino results for widths 8, 12, 16 are also computed and agree with Merino. The stated contribution is computational: explicit generating functions, recurrences, and new sequences, with code available on GitHub.","tokens_in":8776,"tokens_out":13364,"duration_ms":112284,"significance":"If the reported outputs are correct, the paper provides a useful demonstration of Zeilberger's automated enumeration framework and supplies explicit exact results for previously unlisted L-tetromino tiling sequences. The recovery of the known 4×2n L-tetromino result and Merino's T-tetromino sequences gives positive evidence that the pipeline is sound. However, the new results are not yet trustworthy: three of the displayed generating functions have a sign error (constant term −1 instead of 1), three recurrence statements have off-by-one validity bounds, and the k=9 generating function promised in the appendix is missing. These errors are fixable, but they must be corrected and independently checked before the central claims can be accepted.","major_comments":[{"comment":"The displayed generating functions are not the generating functions of the stated sequences. In Proposition 6, A(t) has constant term −1 and B(t) has constant term 1, so A(0)/B(0) = −1, whereas the sequence has a0=1. Expanding the displayed ratio gives −1,−5,... rather than 1,5,34,... . The correct GF is −A(t)/B(t). In Propositions 7 and 8, A(0)=1 and B(0)=−1, so A(0)/B(0)=−1 as well; the correct GF is A(t)/(−B(t)). These are load-bearing sign errors: every displayed new L-tetromino GF in the appendix is wrong as printed.","section":"Appendix A.2, Propositions 6–8"},{"comment":"The stated recurrence bounds contradict the displayed generating functions. For Proposition 2, the GF expands to a4=86, while the recurrence a_n=2a_{n−1}+4a_{n−2}+4a_{n−3}+4a_{n−4} with n≥4 and the given initial values gives a4=88; the correct bound is n≥5. For Proposition 3, n≥2 gives a2=69 but the GF gives a2=84; the correct bound is n≥3. For Proposition 4, n≥4 gives a4=5,249,205 but the GF gives a4=5,253,822; the correct bound is n≥5. The initial-value lists must also include the first term needed for the corrected recurrence.","section":"Propositions 2–4"},{"comment":"Table 3 lists the sequence for 9×8n boards, and the text says 'the corresponding generating functions for each case are listed in Appendix A.2.' However, Appendix A.2 contains Propositions 6–8 only; there is no generating function for k=9. Since this is the largest new width in the paper's main table, the omission is material. Please add the GF or clearly state that it is only in the repository and amend the text accordingly.","section":"Appendix A.2 / Table 3"},{"comment":"The new L-tetromino generating functions are asserted as direct outputs of Zeilberger's Maple package. The only reported checks are recovery of the known 4×2n L-tetromino GF and agreement with Merino's T-tetromino sequences. Given the sign and boundary errors above, these checks are insufficient for the new widths. I request an explicit machine-checkable verification for each new GF and recurrence, e.g., a worksheet that expands the GF to the first 20 coefficients and compares with a direct transfer-matrix or backtracking enumeration for small n, plus a script verifying each recurrence against its GF.","section":"General verification"}],"minor_comments":[{"comment":"Citation error: 'See Elkies et al. [2]' should cite reference [3], not [2].","section":"§2.1"},{"comment":"The generating functions should be given in normalized form with constant term 1 in the denominator; the current sign presentation makes it very easy to misread the outputs.","section":"Propositions 6–8"},{"comment":"The definition of the 'smallest scaling factor' is informal. For k=8 the sequence still contains zero terms, so the criterion needs to be stated more carefully.","section":"Table 3"},{"comment":"For the corrected recurrences, list all initial values actually needed (e.g., Proposition 2 should include a4=86 alongside a0,...,a3).","section":"Propositions 2–4"}],"recommendation":"major_revision","confidential_remarks":"The pattern of errors — off-by-one recurrence bounds and sign errors in three generating functions — suggests the manuscript was not checked against its own formulas. Since the contribution is entirely computational, I strongly recommend requiring the authors to deposit the code and output files, and to include a verification script with the revised submission. The errors are readily fixable, so I support major revision rather than rejection, but the revised version must be free of these inconsistencies."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Zeilberger's 2006 transfer-matrix package does recover the Bělohoubek–Slavík generating function for the 4×2n L-tetromino case, and the paper then applies the same crank to widths 5–9. The width-5 through width-8 generating functions in Appendix A.2 and the sequences in Table 3 are new as far as I can tell, and the T-tetromino outputs match Merino's table. The exposition is clear about what is program output and what is proof, and the code is on GitHub. This is an honest, useful computational data paper.\n\nThe soft spots, in order. First, every displayed recurrence has the same boundary error. Proposition 2 states a_n = 2a_{n−1}+4a_{n−2}+4a_{n−3}+4a_{n−4} for n≥4, but that recurrence with the stated initial values gives a_4 = 88, while the generating function expands to 86. The correct bound is n≥5. The same off-by-one appears in Propositions 3 and 4, and presumably throughout the appendix. This does not invalidate the generating functions — I checked the expansions and the GFs are consistent with their listed first terms — but it means the manuscript was never checked against its own formulas.\n\nSecond, the load-bearing new results, the L-tetromino generating functions for widths 5–9, have no independent verification. The k=4 recovery is a good sanity check, but a state-equation bug for the new widths would silently change every new sequence, and the recurrence errors make it harder to take unverified program output at face value.\n\nThird, the k=9 generating function promised in the abstract, Table 3, and the appendix's own description is absent from Appendix A.2; the largest new sequence has no formula to verify. Minor: the OEIS \"new\" claims lack search details, and the zero-laden k=8 sequence deserves a comment on whether it is a subsequence of something known.\n\nThis paper is for combinatorialists who work on polyomino tilings and want exact sequences for OEIS or bijective work. It is an extension of a known method, not a conceptual advance, but the data is genuinely new, and the benchmark-then-extend approach is right. It deserves peer review, on the condition that the author fixes the recurrence bounds, supplies the k=9 generating function, and adds an independent check (a second implementation or a machine-checked certificate) for the new widths. I would not desk-reject it.","headline":"Zeilberger's package recovers the known k=4 case, then produces genuinely new L-tetromino data for widths 5–9, but every displayed recurrence is off by one at the boundary and the promised k=9 generating function never appears.","tokens_in":9266,"tokens_out":7800,"would_cite":false,"duration_ms":61032,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B45","05A15","11B37"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper demonstrates that an automated transfer-matrix enumeration program yields exact rational generating functions and recurrences for tilings of k×n boards by L-tetrominoes for widths 5 through 9.","keywords":["polyomino tilings","L-tetromino","transfer-matrix method","generating functions","linear recurrences","rectangular boards","rational generating functions","automatic enumeration"],"falsifier":"Compute the first several coefficients of any reported generating function—for instance, the 5×8n sequence 1,2,7,24,86,304,1076,3808—by an independent exhaustive tiling enumeration of the corresponding boards; a single mismatch (such as a 5×16 board admitting other than 7 tilings) would falsify the paper's claim for that width.","tokens_in":8170,"feed_emoji":"🧩","tokens_out":8952,"duration_ms":73087,"temperature":0.7,"pith_summary":"This paper argues that an automated transfer-matrix enumeration program—rather than hand-built case analysis—can count tilings of rectangular boards by L-tetrominoes for several widths at once. The main new results are exact rational generating functions and linear recurrences for the numbers of tilings of k×n boards with k=5,6,7,8,9, allowing rotations but not reflections. For example, the number of tilings of a 5×8n board satisfies a_n = 2a_{n−1}+4a_{n−2}+4a_{n−3}+4a_{n−4}. The paper first checks the method by automatically recovering a recently published generating function for the 4×2n case, then reports sequences for the wider boards that are not currently listed in standard integer-sequence databases. The contribution is a uniform computational route to exact, previously unlisted tiling counts.","feed_headline":"Five new L-tetromino tiling sequences found automatically","feed_subtitle":"Transfer-matrix method yields exact generating functions and recurrences for k×n boards with k=5 through 9.","key_machinery":"The engine is a transfer-matrix construction over a fixed-width board. Each tile is stored as a normalized set of lattice points, and the tiling is built column by column; the state at each step records which cells of the current column are already occupied by tiles protruding from the left. The program assembles a finite linear system whose variables are generating functions for the original board and for auxiliary jagged boards, solves the system symbolically, and reads off a rational generating function and hence a linear recurrence. For L-tetrominoes the tile set consists of the four rotations of the L-shaped tile, and the correctness of the method is first checked against the known 4×2n","core_discovery":"The central claim is that an automated tiling-enumeration framework, which sets up a finite system of linear equations for a rectangular board and its auxiliary jagged boards, solves the system symbolically, and converts rational generating functions into recurrences, correctly enumerates L-tetromino tilings for widths 5 through 9. For 5×8n boards the framework gives the rational generating function (2t^4+2t^3+t^2−1)/(4t^4+4t^3+4t^2+2t−1) and the linear recurrence a_n = 2a_{n−1}+4a_{n−2}+4a_{n−3}+4a_{n−4}, starting from a_0=1, a_1=2, a_2=7, a_3=24. Propositions 6–8 state analogous generating functions for 6×4n, 7×8n, 8×n, and 9×8n boards. The paper verifies the framework by reproducing the k","pith_inferences":["The width-8 L-tetromino sequence, with zeros at odd n, suggests geometric tilability conditions that could be isolated by analyzing when the state system forces zero coefficients; such a divisibility criterion would complement the enumeration results.","An independent brute-force enumerator for small n at each new width would provide a cheap check that the transfer-matrix outputs are exact, since the paper presents the new propositions as program outputs rather than formal proofs.","The same transfer-matrix tool could be run on other tetromino classes under rotations, or on L-tetrominoes with reflections allowed, to see which families yield low-order rational generating functions and which exhibit irregular zero patterns.","For widths beyond 9, the state space grows quickly; a testable extension would be to compare the automatic generating function with coefficients from dynamic programming at moderate n, mapping the practical limits of the method."],"forward_implications":["Exact formulas now exist for L-tetromino tilings of five new board families: the paper supplies generating functions and recurrences for widths 5 through 9, with the 5×8n recurrence explicitly a_n = 2a_{n−1}+4a_{n−2}+4a_{n−3}+4a_{n−4}.","The resulting integer sequences are not currently listed in standard sequence databases, so the paper extends the available enumeration data for polyomino tilings.","Because every generating function is rational with an explicit denominator, each sequence can be generated rapidly to arbitrary length by the corresponding linear recurrence.","The framework is not limited to the L-tetromino: it handles arbitrary tile sets, including reflected tiles and other free polyominoes, and can in principle reach widths up to at least 16.","The auxiliary files accompanying the paper also record asymptotics for the listed sequences, so growth estimates are part of the output."],"fun_headline_variants":["Automated method yields five new L-tetromino sequences","New L-tetromino tilings for widths 5-9 computed automatically","Exact generating functions for L-tetromino tilings up to width 9","Automatic enumeration reveals new L-tetromino tiling counts","Five new L-tetromino tiling sequences from transfer matrices"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is the correctness of the automated transfer-matrix implementation for widths 5 through 9; if the program makes an error in building or solving its state equations at any of these widths, every new formula and sequence in the paper would be wrong.","fun_headline_variants_meta":{"raw":{"variants":["Automated method yields five new L-tetromino sequences","New L-tetromino tilings for widths 5-9 computed automatically","Exact generating functions for L-tetromino tilings up to width 9","Automatic enumeration reveals new L-tetromino tiling counts","Five new L-tetromino tiling sequences from transfer matrices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000231,"raw_usage":{"total_tokens":1310,"prompt_tokens":719,"completion_tokens":591,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":494}},"tokens_in":463,"tokens_out":591,"duration_ms":5599,"temperature":1.0,"reasoning_tokens":494,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T08:21:45.868975+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the first several coefficients of any reported generating function—for instance, the 5×8n sequence 1,2,7,24,86,304,1076,3808—by an independent exhaustive tiling enumeration of the corresponding boards; a single mismatch (such as a 5×16 board admitting other than 7 tilings) would falsify the paper's claim for that width.","supporting_citations":[],"review_version":1}