A331450 Irregular triangle read by rows: Take a regular n-sided polygon (n>=3) with all diagonals drawn, as in A007678. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., max_k.

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, 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, 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

Offset: 3

Author

Scott R. Shannon and N. J. A. Sloane, Jan 25 2020

Keywords

• nonn
• tabf

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)

Examples

			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,
  ...
The row sums are A007678, the first column is A062361.
		

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].
• N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]
• 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]

Crossrefs

Cf. A007678, A062361, A331451, A341729, A341730, A335000.

Extensions

Added "regular" to definition. - N. J. A. Sloane, Mar 06 2021