math.CO — Combinatorics
Tilings of an equilateral triangle by at most five lattice trapezoids with 60° base angles: a complete structural classification
We study tilings of an equilateral triangle of side n in the triangular grid by k lattice trapezoids with base angles 60°, the objects behind the OEIS sequences A389392 (k = 4) and A391498 (k = 5). We prove two angle identities valid for every such tiling and a lemma relating the number of boundary vertices to the number of pieces having a side on the boundary. With these tools we show that there are exactly 1, 2 and 13 combinatorial types of tilings for k = 3, 4, 5. For k = 3 every tiling is a pinwheel. For k = 4 every tiling belongs to one of the two categories used in A389392, a fact that had previously been taken for granted. For k = 5 the thirteen types refine the eight categories of A391498; in three of them a piece has no side on the boundary. As a consistency check, the volumes of the parameter polytopes together with the generic multiplicities reproduce the leading coefficient 7/36 of the conjectured quasi-polynomial for A391498. All lemmas were checked against an exhaustive enumeration of the tilings with pairwise distinct pieces for n ≤ 15. Finally, the classification turns the thirteen types into eight explicit families of sets of shapes, and an inclusion–exclusion over them reduces the conjectured generating function of A391498 to twelve elementary counting statements, and we settle all twelve: the resulting closed formula reproduces the sequence for every n ≤ 127.