OFFSET
3,2
COMMENTS
Computed by Scott R. Shannon, Jan 24 2020
See A331451 for a version of this triangle giving the counts for k = 3 through n.
Mitosis of convex polygons, by Scott R. Shannon and N. J. A. Sloane, Dec 13 2021 (Start)
Borrowing a term from biology, we can think of this process as the "mitosis" of a regular polygon. Row 6 of this triangle shows that a regular hexagon "mitoses" into 18 triangles and 4 quadrilaterals, which we denote by 3^18 4^6.
What if we start with a convex but not necessarily regular n-gon? Let M(n) denote the number of different decompositions into cells that can be obtained. For n = 3, 4, and 5 there is only one possibility. For n = 6 there are two possibilities, 3^18 4^6 and 3^19 4^3 5^3. For n = 7 there are at least 11 possibilities. So the sequence M(n) for n >= 3 begins 1, 1, 1, 2, >=11, ...
The links below give further information. See also A350000. (End)
LINKS
M. Rubinstein, Drawings of A007678 for n=4,5,6,....
Scott R. Shannon, Rows 3 through 596 of A331450.
N. J. A. Sloane, Illustration for row n=9 of A331450. [9-gon with one representative for each type of polygonal cell labeled with its number of sides].
Scott R. Shannon and N. J. A. Sloane, N. J. A. Sloane, Table showing number of ways to "mitose" a convex n-gon. [From here on the links are related to the mitosis problem, and are in logical rather than alphabetical order]
Scott R. Shannon, Illustration for mitosis 5.1 of a pentagon (the cell counts are the same whether of not the pentagon is regular)
Scott R. Shannon, Illustration for mitosis 6.1 of a hexagon with a triple point (the cell counts are the same whether of not the hexagon is regular, as long as it has a triple point)
Scott R. Shannon, Illustration for mitosis 6.2 of a hexagon (without a triple point)
N. J. A. Sloane, Illustration for mitosis 7a of a 7-gon
Scott R. Shannon, A second (colored) illustration for mitosis 7a of a 7-gon
Scott R. Shannon, Illustration for mitosis 7b
Scott R. Shannon, Illustration for mitosis 7c
Scott R. Shannon, Illustration for mitosis 7d
Scott R. Shannon, Illustration for mitosis 7e
Scott R. Shannon, Illustration for mitosis 7f
Scott R. Shannon, Illustration for mitosis 7g
Scott R. Shannon, Illustration 1 for mitosis 7h
Scott R. Shannon, Illustration 2 for mitosis 7h (same cell counts as preceding illustration but a different polygon - look at the brown cells)
Scott R. Shannon, Illustration for mitosis 7i
Scott R. Shannon, Illustration for mitosis 7j
Scott R. Shannon, Illustration for mitosis 7k
M. F. Hasler, Nine examples of dissections of convex 7-gons [These are all subsumed in the above illustrations]
EXAMPLE
A hexagon with all diagonals drawn contains 18 triangles and 6 quadrilaterals, so row 6 is [18, 6].
Triangle begins:
1,
4,
10, 0, 1,
18, 6,
35, 7, 7, 0, 1,
56, 24,
90, 36, 18, 9, 0, 0, 1,
120, 90, 10,
176, 132, 44, 22, 0, 0, 0, 0, 1
276, 168,
377, 234, 117, 39, 0, 13, 0, 0, 0, 0, 1,
476, 378, 98,
585, 600, 150, 105, 15, 0, 0, 0, 0, 0, 0, 0, 1,
848, 672, 128, 48,
1054, 901, 357, 136, 17, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1,
1404, 954, 72, 18, 18,
1653, 1444, 646, 190, 57, 38, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
2200, 1580, 580, 120, 0, 20,
2268, 2520, 903, 462, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
2992, 2860, 814, 66, 44, 44,
3749, 2990, 1564, 644, 115, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
...
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Scott R. Shannon and N. J. A. Sloane, Jan 25 2020
EXTENSIONS
Added "regular" to definition. - N. J. A. Sloane, Mar 06 2021
STATUS
approved