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

3,2

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

Table of n, a(n) for n=3..114.

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

Crossrefs

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

Sequence in context: A111443 A299118 A276272 * A051499 A333654 A008345

Adjacent sequences: A331447 A331448 A331449 * A331451 A331452 A331453

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