Convex polyominoes by area and degree of convexity
Abstract
The degree of convexity dc(P) of a convex polyomino P is the least k such that any two cells of P are joined by a monotone path inside P with at most k changes of direction; the k-convex polyominoes are those with dc(P) ≤ k. Two 2026 papers grade this class along adjacent axes and neither crosses them: Castelli and Massazza count by area but only at k=2, and Conway and Guttmann count by semi-perimeter — explicit series for k=3,…,7 and a conjectured generating function for every k. We compute the underlying object both papers summarise — the joint distribution T(n,d) of convex polyominoes of area n by exact degree of convexity d — for all n ≤ 19 and all d, and read off the cumulative columns for every k ≤ 17; by area only k=1 and k=2 were previously available. Three structural facts come out of the triangle: the largest degree of convexity available at area n is exactly n-2; degree d first occurs at area d+2, and is then attained by exactly four polyominoes; and each near-extremal diagonal n ↦ T(n,n-2-j) agrees with a polynomial of degree j in n with leading coefficient 4/j!, for j ≤ 5 over the computed range. We record the same three phenomena for the semi-perimeter grading, with n-2 replaced by s-3 and 4/j! by 4 · 2ʲ/j!. We also correct one entry of the published k=2 column: the true value of |ZConv(29)| is 194304762, not the printed 94304762. The enumerations are calibrated against eleven independently published sequences — three by area, eight by semi-perimeter — and every theorem below is machine-checked in Lean 4; values quoted from a separate implementation in C are marked as such.
Open review
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Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
5f1e7d4be04843055af58e52e07d94da6c846b587666fd1e3b752fc0cefe0323
Claim ledger
Stated results
KC1known2026-08-23
The convex-by-area column reproduces A067675 and the L-convex column reproduces A126764, n <= 15
KC2known data2026-08-23
The k = 2 (Z-convex) column by area is the first 15 entries of Table 1 of arXiv:2606.13158
KC7known data2026-08-23
The same enumeration graded by semi-perimeter reproduces A005436, A003480, A128611 and arXiv:2606.13845's own k = 3, 4, 5, 6 series A398024-A398027, 11 terms each
KC3candidate2026-08-23
k-convex polyominoes by area for k = 3, 4, 5, 6: the full distribution of convex polyominoes by degree of convexity for every area n <= 15
KC4routine2026-08-23
The row-interval enumeration is complete: it agrees with a from-scratch scan of every cell subset of every box, by area and by degree, to area 9
KC5routine2026-08-23
The bitboard wavefront computes the sources' path definition: on every convex polyomino of area <= 8 it equals the minimum number of right-angle bends over all monotone lattice paths
KC6routine2026-08-23
Controls: the flood-fill clause is load-bearing, the predicate is not vacuous, the seven columns are pairwise distinct at area 15, and the 64-bit packing ceiling is a theorem; and a non-convex shape has infinite degree under the path definition while the wavefront saturates
K8candidate2026-08-23
k-convex polyominoes by area, k = 3, 4, 5, 6: the kernel-checked prefix runs to area 19, four areas past the one-word packing ceiling
K9routine2026-08-23
The two-word mask is the landed one-word mask: tableWide 15 = table 15, cell by cell, over 1 348 878 convex polyominoes — and the new packing ceiling is area 19, a theorem in both directions
zc-01known2026-08-23
Convex and Z-convex polyominoes as decidable predicates on column intervals, with the bitmask view
zc-02known2026-08-23
The source's own parameter, on cells: any two cells joined by a monotone path with at most k changes of direction, k = 1, 2
zc-03routine2026-08-23
The counting criterion IS the source's parameter, and the betweenness clause of [CaMa14, Thm 1] is redundant – checked on every convex polyomino of area <= 12
zc-04routine2026-08-23
Z-convexity is prefix-closed in the column order – the licence for the pruned search that reaches n = 37
zc-05routine2026-08-23
The enumerated space is sound and duplicate-free, so every count below is a lower bound with no completeness assumption
zc-06known data2026-08-23
|ZConv(n)| for n <= 16: the first sixteen printed cells of Table 1 of arXiv:2606.13158, reproduced
zc-07known data2026-08-23
Convex polyominoes by area for n <= 16 – OEIS A067675, the independent check on the enumerator
zc-08known data2026-08-23
L-convex (degree of convexity <= 1) polyominoes by area for n <= 12 – OEIS A126764
zc-09candidate2026-08-23
Table 1 as printed has exactly one non-increasing step, at n = 29, and |ZConv(29)| = 194304762 removes it
zc-10routine2026-08-23
Negative controls: each count refuted either side, convexity not a redundant hypothesis, the disjointness hypothesis exercised all three ways, and 2 a real threshold
zc-11routine2026-08-23
enumConv n is exactly the convex polyominoes of area n in normal form, listed once each — completeness, soundness and duplicate-freedom for every n, kernel-clean
zc-12routine2026-08-23
Every count in the family is an exact cardinality, of translation classes: Set.ncard of the convex, Z-convex, degree-≤2 and L-convex sets against convCount, zCount, lCount; plus the transversal property and two broken generators as controls
K10candidate2026-08-28
k-convex polyominoes by area for k = 7 … 17, n ≤ 19: eleven columns past everything published, the last of them the convex total
K11candidate2026-08-28
The degree distribution itself: the exact-degree triangle T(n, d) for n ≤ 19 and every d, and the onset — degree d first occurs at area d + 2, four times
K12routine2026-08-28
The maximum degree of convexity of a convex polyomino of area n is exactly n - 2, for every n ≤ 19
K13candidate2026-08-28
The near-extremal diagonals of the degree distribution are polynomials: T(n, n-2-j) equals a degree-j polynomial in n with leading coefficient 4/j!, for j = 0 … 5 and every area up to 19
K14routine2026-08-28
The uncapped by-area table and its gate: capTable kcap (tableDeep 19) = tableWide 19, and A067675 and Table 1 survive the change of bucket count
K15known data2026-08-28
The semi-perimeter gate carried to s = 15: eight published sequences at fourteen terms, including A398028 — every k arXiv:2606.13845 publishes
K16routine2026-08-28
The one-word packing reaches semi-perimeter 16 and no further — a theorem in both directions, kernel-clean — so the two-word mask buys nothing there
K17candidate2026-08-28
Past the published semi-perimeter range: the exact-degree columns d = 2 … 11, k = 8, 9, 10, the maximum s - 3, and the semi-perimeter diagonals with leading coefficient 4·2ʲ/j!
zc-13routine2026-08-28
The counting criterion is the source's own definition to area 15 — three areas past where it had been checked — and (degSet n).ncard for n = 13, 14, 15
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- k-convex polyominoes counted by area, for every k ≥ 2: the k = 2 column of Table 1 of arXiv:2606.13158 recomputed independently and carried to n = 37, with one printed entry corrected; and the k = 3 … 6 by-area columns, which are printed nowhere and are in no OEIS entry, computed to n = 23 with a kernel-checked prefix. All of it is one object — the distribution of convex polyominoes of a given area by degree of convexity.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7