{"id":"54308477-2b6e-4992-bef3-a76da3b2176e","arxiv_id":"2411.17812","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper gives rational generating functions for p-Fibonacci polyominoes by area, semiperimeter, and inner points, plus bijections to binary words and compositions.","lead":"This paper defines a new family of shapes called p-Fibonacci polyominoes, which are column charts built from words counted by generalized Fibonacci numbers, and derives exact formulas for their area, semiperimeter, and inner points. The paper matters as a worked example of how generating functions and a simple geometric reading of a known word family yield exact enumeration for a new shape class.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central recurrences rest on an unstated semiperimeter/inner-point increment lemma; I checked it and it is true, so the main formulas stand but the proof gap and the misleading §2.1 decomposition justify keeping the conditional verdict.","rationale":"I find the same load-bearing gap the reader identified: the recurrences in Theorems 2.1 and 2.3 depend on unproved semiperimeter and inner-point increment rules. I checked the rules against the perimeter formula and Pick's theorem; they are correct, and the p = 2, 3, 4 expansions match direct enumeration, so the central generating functions are not false. The concern is therefore that the proof as written is incomplete: the geometric lemma should be stated and proved, and §2.1's verbal decomposition should be corrected to a rightmost-append description. I also note that the reverse direction of Theorem 2.4 swaps 0 and 1 in its construction (the example 1011011 → 33233233 corresponds to the opposite convention from the stated 0 = increase/unchanged, 1 = decrease), but this is a fixable misstatement in a secondary bijection, not a threat to Theorem 2.1. For these reasons the reader's CONDITIONAL verdict is appropriate; I would not raise it to ACCEPT until the lemma and the decomposition are fixed.","tokens_in":119,"tokens_out":21453,"duration_ms":486126,"concrete_test":"Independently compute the increment deltas: for heights j then i, Δsper = 1 + (i − j)_+, Δarea = i, so by Pick's theorem Δinn = i − (1 + (i − j)_+); then verify these exponents occur in Eq. (5) and in the F_{p,p}/G_{p,p} recurrences. As an additional check, enumerate all p-Fibonacci words for p = 3, 4 and n ≤ 4, compute area and semiperimeter by the explicit perimeter formula and inner points by Pick's theorem, and compare the weighted sums to the coefficients of x^n in Theorems 2.1 and 2.3; all coefficients should coincide.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.2, Eq. (5) and the equation just before 'Combining these results' define the recurrences used to prove Theorem 2.1, but the z-exponents encode a geometric rule that is never stated: appending a column of height i to the right of a column of height j increases semiperimeter by 1 if i ≤ j and by 1 + (i − j) if i > j. Similarly, Theorem 2.3's recurrences G_{p,i} = x q^{i−1} G_{p,i+1} and G_{p,p} = x + Σ x q^{j−1} G_{p,j} assume that appending a column of height i changes the number of inner points by i − 1, via Pick's theorem. If these increments were wrong, Theorems 2.1 and 2.3 would be wrong; the paper never proves them. Worse, the verbal decomposition in §2.1 says 'P starts with a column of p cells, followed by P′', while the recurrences are rightmost appends, so a reader cannot validate the exponents from the stated decomposition. Direct derivation from sper = n + (h1 + hn)/2 + ½Σ|h_i − h_{i+1}| and area = inn + sper − 1 confirms every exponent, and the p = 2, 3, 4 series match, so this is a rigor/exposition gap rather than a mathematical counterexample.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines p-Fibonacci words and the bargraph polyominoes obtained by taking the symbols as column heights. The main results are multivariate generating functions for p-Fibonacci polyominoes with respect to length, area, semiperimeter, and number of inner points (Theorems 2.1 and 2.3), closed forms for the total area, semiperimeter, and inner points over all polyominoes with n columns, a generating function for polyominoes of a given area together with a bijection to compositions with parts in a finite set A_p (Theorem 2.2), and a claimed bijection between p-Fibonacci words and binary words avoiding p consecutive 1s (Theorem 2.4). The paper also connects the three statistics via Pick's theorem through the identity F_{p,n+1}=ip(n)+sp(n)-ap(n).","tokens_in":10406,"tokens_out":10820,"duration_ms":95373,"significance":"If the proofs are brought to the same standard as the formulas, the paper gives exact, parameter-free enumerations of a natural class of bargraph polyominoes by several geometric statistics, with rational generating functions that can be checked against small cases and differentiated to obtain total statistics. The composition bijection in Theorem 2.2 is elegant and appears correct. The main weakness is not the validity of the formulas but the presentation of the underlying geometric recurrences and one internally inconsistent bijection proof; both are repairable. As it stands, the manuscript is a useful contribution but needs revision before it can be accepted.","major_comments":[{"comment":"The recurrences used in Theorems 2.1 and 2.3 encode a geometric increment rule that is never stated or proved. For a bargraph with last column height j, appending a column of height i to the right changes the semiperimeter by 1 if i ≤ j and by 1+(i−j) if i > j; this is what the z-exponents in Eq. (5) and in the equation for F_{p,p} express. Similarly, the recurrences G_{p,i}=x q^{i−1}G_{p,i+1} and G_{p,p}=x+Σ_{j=1}^p x q^{j−1}G_{p,j} assume, via Pick's theorem, that appending a column of height i changes the number of inner points by i−1 (or j−1 in the p-append case). I checked the rule directly from sper = n + (h_1+h_n)/2 + ½Σ|h_i−h_{i+1}| and from area = inn + sper − 1, and it is correct, so the formulas are not invalidated. Nevertheless, the proof of Theorem 2.1 is incomplete as written because the exponents are asserted without geometric justification, and a reader cannot verify them from the stated decomposition.","section":"§2.1, Cases 1 and 2"},{"comment":"The verbal decomposition in §2.1 is incompatible with the recurrences. The text says a polyomino in B^{(p,i)}_n 'starts with a column of p cells, followed by' a polyomino P′ in B^{(p,i+1)}_{n−1}, and similarly for Case 2. But the recurrences F_{p,i}=xy^i z F_{p,i+1} and F_{p,p}=xy^p z^{p+1}+xy^p Σ_j z^{p+1-j}F_{p,j} are rightmost-append recurrences: they remove or add the last column. As written, if P starts with p and is followed by a polyomino ending in i+1, then P ends in i+1, not i, so the case division does not match the definitions of B^{(p,i)}_n. The proof should be rewritten in terms of appending a column to the right of a polyomino whose last column has a specified height, and the geometric increment lemma above should be stated and proved there.","section":"§2.4, Theorem 2.4"},{"comment":"The proof of Theorem 2.4 is internally inconsistent. The forward map assigns 0 to increases or unchanged digits and 1 to decreases. The reverse construction, however, says to append the 'allowed descending digit' on a 0 and the 'allowed ascending or unchanged digit' on a 1, which is exactly the opposite convention. Under the stated forward map, the example word 33233233 has transitions 3→3, 3→2, 2→3, 3→3, 3→2, 2→3, 3→3, so it maps to 0100100, not to 1011011. Conversely, 1011011 generates 33233233 only if 0 is interpreted as a descent and 1 as an ascent. The sentence 'each word has at most p−1 ascents' should also read 'at most p−1 consecutive descents.' The bijection itself is plausible and can be repaired by fixing the inverse rule and the example, but the proof as written does not establish the theorem.","section":"Conclusion, Figure 8"}],"minor_comments":[{"comment":"The displayed set W^{(3)}_4 contains the word 3323 twice; there are only seven 3-Fibonacci words of length 4, as shown correctly in Figure 2.","section":"§2.3.1"},{"comment":"The subsection title contains a typo: 'Fibonaci' should be 'Fibonacci.'","section":"§2.4 closing paragraph"},{"comment":"The sentence 'the subset of p-generalized Fibonacci words of length n or equivalently the p-generalized Fibonacci polyominoes with area n' conflates length (number of columns) with area; a word of length n corresponds to a polyomino with n columns, not area n.","section":"Figure 8"},{"comment":"The examples in Figure 8 do not appear to follow the bijections stated in Theorem 2.4: for instance, the word 32321 would correspond to the binary word 1011 under the mapping of Theorem 2.4, not to 110. The figure should be redrawn or its conventions explained.","section":"Theorem 2.2 proof"},{"comment":"In the example in the proof of Theorem 2.2, the word 32321 is split into blocks 32 and 321, which is correct, but the sentence introducing the blocks should make clear that the decomposition is deterministic and reads the word from left to right, starting a new block at each occurrence of p that is not part of the current descending run.","section":"Section 2, Eq. (3)"},{"comment":"The identity F_{p,n+1}=ip(n)+sp(n)-ap(n) is presented as 'interesting,' but it follows immediately from Pick's theorem after summing over the F_{p,n+1} polyominoes; this derivation should be stated explicitly in one sentence.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the scope of math.CO and the central enumerative formulas appear to be correct; the issues are proof presentation and an inconsistent bijection proof. The bijection error in Theorem 2.4 is localized and fixable, so I do not recommend rejection. I would ask the authors to add the explicit semiperimeter/inner-point increment lemma, rewrite the decomposition in §2.1 as a rightmost-append argument, and correct Theorem 2.4's inverse rule and example. The duplicate word in W^{(3)}_4 and the confusing Figure 8 should also be fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main formulas hold up. I checked the recurrences in Theorems 2.1 and 2.3 against the geometry, and the small cases match. This is a genuine extension of Kirgizov and Ramirez's p=2 work to p-generalized Fibonacci polyominoes, with new generating functions for area, semiperimeter, and inner points, plus a Pick's-theorem identity that gives F_{p,n+1} = ip(n) + sp(n) - ap(n). That identity is a nice sanity check and the paper earns credit for it.\n\nThe soft spots are real but not fatal. The recurrences in Section 2.2 and 2.3 depend on an implicit geometric increment rule: appending a column of height i to a column of height j changes semiperimeter by 1 if i<=j and by 1+(i-j) if i>j, and inner points by i-1 in the drop case. The rule is true, and it follows from the standard formula for semiperimeter of a bargraph, but the paper never states or proves it, so the proofs of Theorems 2.1 and 2.3 are gaps as written. The verbal decomposition in Section 2.1 is also misleading: it says 'P starts with a column of p cells' while the recurrences are rightmost appends, so a reader cannot easily verify the exponents. That should be rewritten.\n\nThe concrete error is in the example of Theorem 2.4. Under the stated rule (0 for increase/unchanged, 1 for decrease), the binary word 1011011 maps to 0100100, not to 33233233. The bijection itself seems plausible, but the example needs to be corrected. There is also a duplicate word 3323 in W^{(3)}_4 on the first page, which is a minor typo.\n\nThese are exactly the kind of issues a referee would catch in revision. The mathematics is sound, the formulas are checkable, and the citation pattern is fine. I would send it to a serious referee rather than desk-reject it. This is for enumerative combinatorists working on word-induced polyominoes; they will find it useful, and after a revision fixing the increment lemma and the example, it would be a solid publication.","headline":"Correct generating functions for a real generalization of Fibonacci polyominoes to p>2, but the paper has a few fixable exposition errors, including a wrong example in Theorem 2.4.","tokens_in":10979,"tokens_out":2612,"would_cite":true,"duration_ms":23342,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A15","05A05","11B39"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper derives exact rational generating functions that enumerate p-Fibonacci polyominoes by length, area, and semiperimeter, and by inner points, with bijections to restricted compositions and binary words.","keywords":["p-Fibonacci word","polyomino","bargraph","generating function","p-generalized Fibonacci numbers","semiperimeter","inner points","integer compositions"],"falsifier":"Take $p=3$, enumerate all 3-Fibonacci words of lengths $1$ through $4$, build the corresponding bargraphs, compute area, semiperimeter, and inner points directly, and compare the resulting coefficient lists with the series expansions of $F_3(x;y,z)$ and $G_3(x;q)$ printed in the paper; any mismatch in a coefficient or exponent would falsify the formulas.","tokens_in":9904,"feed_emoji":"🧩","tokens_out":13109,"duration_ms":116894,"temperature":0.7,"pith_summary":"The paper studies p-Fibonacci polyominoes, bargraphs whose column heights form p-Fibonacci words, and establishes exact enumerations by length, area, semiperimeter, and inner points. Its main result is a rational multivariate generating function that packages all three shape statistics, from which closed forms for total area and total semiperimeter follow by differentiation. A second generating function handles inner points, and an identity from Pick's theorem connects the three statistics to the p-generalized Fibonacci numbers. The paper also proves bijections between these polyominoes, restricted compositions, and binary words avoiding $p$ consecutive ones. If the formulas are correct, these otherwise complicated objects become exactly and computationally tractable.","feed_headline":"Rational formula counts all p-Fibonacci polyominoes","feed_subtitle":"Closed generating functions give total area, semiperimeter, and inner points, plus bijections to compositions and binary words.","key_machinery":"The engine is a last-column recurrence. In a p-Fibonacci word, a symbol $i<p$ must be preceded by $i+1$, while $p$ may follow any symbol; this makes the class recursive and produces a finite system of equations for the generating functions $F_{p,i}$ and $G_{p,i}$. Solving those systems gives Theorem 2.1 and Theorem 2.3. The same structure appears as the generating tree $\\Omega_p$, and reading a word as concatenated descending blocks $p,p-1,\\dots,p-k$ yields the bijection to compositions with parts $(p+i)(p-i+1)/2$.","core_discovery":"The central claim is that the class of p-Fibonacci polyominoes, bargraphs whose column heights form a p-Fibonacci word, is exactly enumerated by the rational generating function $F_p(x;y,z)=1+\\frac{\\sum_{i=1}^p x^{p-i+1}y^{(p+i)(p-i+1)/2}z^{2p-i+1}}{1-xy^p z-\\sum_{i=1}^{p-1}x^{p-i+1}y^{(p+i)(p-i+1)/2}z^{2p-2i+1}}$, where $x$ marks columns, $y$ marks area, and $z$ marks semiperimeter. A parallel generating function $G_p(x;q)$ counts the same polyominoes by inner points. From these, the paper obtains closed rational forms for the total area, total semiperimeter, and total inner points over all polyominoes with $n$ columns, and derives the identity $F_{p,n+1}=ip(n)+sp(n)-ap(n)$ from Pick's theorem.","pith_inferences":["Because both generating functions are rational, all fixed-order cumulants of area, semiperimeter, and inner points, not just totals, can be obtained by differentiation, and the asymptotic growth of every statistic is controlled by the dominant singularity of $1-x-\\cdots-x^p$; the paper does not pursue these consequences.","The bijection with binary words suggests a one-bit-per-step encoding of p-Fibonacci polyominoes, which would give a simple uniform random generation algorithm and a linear-time decoder from binary code to polyomino.","The last-column recurrence should transfer to other word-defined polyomino families whose production rules give most symbols a unique predecessor, yielding analogous rational generating functions; this generalization is implicit but not developed.","The identity $F_{p,n+1}=ip(n)+sp(n)-ap(n)$ invites a bijective proof: if the expression can be interpreted as counting configurations, it would give a direct combinatorial proof of the p-generalized Fibonacci recurrence."],"forward_implications":["For each fixed $p$, expanding $F_p(x;y,z)$ and $G_p(x;q)$ yields tables of p-Fibonacci polyominoes by length, area, semiperimeter, and inner points, as the paper displays for $p=2,3,4$.","Setting $y=z=1$ in Theorem 2.1 gives $F_p(x;1,1)=1/(1-x-x^2-\\cdots-x^p)$, so the number of polyominoes with $n$ columns is exactly the p-generalized Fibonacci number $F_{p,n+1}$.","Differentiating at $y=z=1$ produces the closed forms for the total area $A_p(x)$ and total semiperimeter $S_p(x)$; differentiating $G_p$ at $q=1$ produces the total inner points $I_p(x)$.","Pick's theorem gives $F_{p,n+1}=ip(n)+sp(n)-ap(n)$, expressing the p-generalized Fibonacci numbers directly through the three geometric statistics.","Theorem 2.2 and Theorem 2.4 give bijections between area-$n$ polyominoes and restricted compositions, and between length-$n$ words and binary words with no $p$ consecutive ones."],"supporting_citations":[{"why":"Proves that the production system $\\Omega_p$ generates the p-generalized Fibonacci sequence, the basis for defining p-Fibonacci words.","marker":"[2]"},{"why":"Supplies the generating-tree methodology used to encode p-Fibonacci words by levels.","marker":"[1]"},{"why":"Shows that binary words avoiding $p$ consecutive ones are counted by p-generalized Fibonacci numbers, the target of Theorem 2.4's bijection.","marker":"[4]"},{"why":"Pick's theorem relates area, boundary lattice points, and interior points, yielding identity (3).","marker":"[12]"},{"why":"Earlier construction of polyominoes from Fibonacci words that the paper generalizes to p-Fibonacci polyominoes.","marker":"[10]"}],"fun_headline_variants":["Closed formulas enumerate p-Fibonacci polyominoes","Rational GF counts all p-Fibonacci polyominoes","Exact area and perimeter sums for p-Fibonacci polyominoes","Bijections map Fibonacci words to polyominoes","Pick's theorem yields identity for p-Fibonacci polyominoes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The recurrences for semiperimeter and inner points rely on an unstated rule for how these statistics change when a column is appended to a bargraph; if that increment rule is wrong, the exponents in Theorems 2.1 and 2.3 fail.","fun_headline_variants_meta":{"raw":{"variants":["Closed formulas enumerate p-Fibonacci polyominoes","Rational GF counts all p-Fibonacci polyominoes","Exact area and perimeter sums for p-Fibonacci polyominoes","Bijections map Fibonacci words to polyominoes","Pick's theorem yields identity for p-Fibonacci polyominoes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000568,"raw_usage":{"total_tokens":2629,"prompt_tokens":826,"completion_tokens":1803,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":442,"completion_tokens_details":{"reasoning_tokens":1724}},"tokens_in":442,"tokens_out":1803,"duration_ms":13418,"temperature":1.0,"reasoning_tokens":1724,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:49:28.919759+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $p=3$, enumerate all 3-Fibonacci words of lengths $1$ through $4$, build the corresponding bargraphs, compute area, semiperimeter, and inner points directly, and compare the resulting coefficient lists with the series expansions of $F_3(x;y,z)$ and $G_3(x;q)$ printed in the paper; any mismatch in a coefficient or exponent would falsify the formulas.","supporting_citations":[{"cited_title":"Baril and P.-T","cited_arxiv_id":null,"evidence_quote":"Proves that the production system $\\Omega_p$ generates the p-generalized Fibonacci sequence, the basis for defining p-Fibonacci words."},{"cited_title":"Barcucci, A","cited_arxiv_id":null,"evidence_quote":"Supplies the generating-tree methodology used to encode p-Fibonacci words by levels."},{"cited_title":"Bernini, Restricted binary strings and generalized Fibonacci numbe rs, In: International Workshop on Cellular Automata and Discrete Complex Systems, Springer (2017 ), 32–43","cited_arxiv_id":null,"evidence_quote":"Shows that binary words avoiding $p$ consecutive ones are counted by p-generalized Fibonacci numbers, the target of Theorem 2.4's bijection."},{"cited_title":"Pick, Geometrisches zur Zahlentheorie , Sitzenber 19 (1899), 311–319","cited_arxiv_id":null,"evidence_quote":"Pick's theorem relates area, boundary lattice points, and interior points, yielding identity (3)."},{"cited_title":"Kirgizov and J.L","cited_arxiv_id":null,"evidence_quote":"Earlier construction of polyominoes from Fibonacci words that the paper generalizes to p-Fibonacci polyominoes."}],"review_version":1}