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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.
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%I #179 Dec 19 2021 17:51:50

%S 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,

%T 44,22,276,168,377,234,117,39,0,13,0,0,0,0,1,476,378,98,585,600,150,

%U 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

%N 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.

%C Computed by _Scott R. Shannon_, Jan 24 2020

%C See A331451 for a version of this triangle giving the counts for k = 3 through n.

%C Mitosis of convex polygons, by _Scott R. Shannon_ and _N. J. A. Sloane_, Dec 13 2021 (Start)

%C 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.

%C 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, ...

%C The links below give further information. See also A350000. (End)

%H M. Rubinstein, <a href="/A006561/a006561_3.pdf">Drawings of A007678 for n=4,5,6,...</a>.

%H Scott R. Shannon, <a href="/A331450/a331450_1.txt">Rows 3 through 596 of A331450</a>.

%H N. J. A. Sloane, <a href="/A331451/a331451.pdf">Illustration for row n=9 of A331450</a>. [9-gon with one representative for each type of polygonal cell labeled with its number of sides].

%H Scott R. Shannon and N. J. A. Sloane, N. J. A. Sloane, <a href="/A331450/a331450_8.txt">Table showing number of ways to "mitose" a convex n-gon.</a> [From here on the links are related to the mitosis problem, and are in logical rather than alphabetical order]

%H Scott R. Shannon, <a href="/A331450/a331450.gif">Illustration for mitosis 5.1 of a pentagon</a> (the cell counts are the same whether of not the pentagon is regular)

%H Scott R. Shannon, <a href="/A331450/a331450_1.gif">Illustration for mitosis 6.1 of a hexagon with a triple point</a> (the cell counts are the same whether of not the hexagon is regular, as long as it has a triple point)

%H Scott R. Shannon, <a href="/A331450/a331450_2.gif">Illustration for mitosis 6.2 of a hexagon</a> (without a triple point)

%H N. J. A. Sloane, <a href="/A331450/a331450_5.pdf">Illustration for mitosis 7a of a 7-gon</a>

%H Scott R. Shannon, <a href="/A331450/a331450_6.gif">A second (colored) illustration for mitosis 7a of a 7-gon</a>

%H Scott R. Shannon, <a href="/A331450/a331450_7.gif">Illustration for mitosis 7b</a>

%H Scott R. Shannon, <a href="/A331450/a331450_18.gif">Illustration for mitosis 7c</a>

%H Scott R. Shannon, <a href="/A331450/a331450_9.gif">Illustration for mitosis 7d</a>

%H Scott R. Shannon, <a href="/A331450/a331450_10.gif">Illustration for mitosis 7e</a>

%H Scott R. Shannon, <a href="/A331450/a331450_11.gif">Illustration for mitosis 7f</a>

%H Scott R. Shannon, <a href="/A331450/a331450_12.gif">Illustration for mitosis 7g</a>

%H Scott R. Shannon, <a href="/A331450/a331450_13.gif">Illustration 1 for mitosis 7h</a>

%H Scott R. Shannon, <a href="/A331450/a331450_14.gif">Illustration 2 for mitosis 7h (same cell counts as preceding illustration but a different polygon - look at the brown cells)</a>

%H Scott R. Shannon, <a href="/A331450/a331450_15.gif">Illustration for mitosis 7i</a>

%H Scott R. Shannon, <a href="/A331450/a331450_16.gif">Illustration for mitosis 7j</a>

%H Scott R. Shannon, <a href="/A331450/a331450_17.gif">Illustration for mitosis 7k</a>

%H M. F. Hasler, <a href="https://drive.google.com/file/d/1PSX2vvpYdmTNlhG9OPdeyAXPTkil_6aH/view">Nine examples of dissections of convex 7-gons</a> [These are all subsumed in the above illustrations]

%e A hexagon with all diagonals drawn contains 18 triangles and 6 quadrilaterals, so row 6 is [18, 6].

%e Triangle begins:

%e 1,

%e 4,

%e 10, 0, 1,

%e 18, 6,

%e 35, 7, 7, 0, 1,

%e 56, 24,

%e 90, 36, 18, 9, 0, 0, 1,

%e 120, 90, 10,

%e 176, 132, 44, 22, 0, 0, 0, 0, 1

%e 276, 168,

%e 377, 234, 117, 39, 0, 13, 0, 0, 0, 0, 1,

%e 476, 378, 98,

%e 585, 600, 150, 105, 15, 0, 0, 0, 0, 0, 0, 0, 1,

%e 848, 672, 128, 48,

%e 1054, 901, 357, 136, 17, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1,

%e 1404, 954, 72, 18, 18,

%e 1653, 1444, 646, 190, 57, 38, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

%e 2200, 1580, 580, 120, 0, 20,

%e 2268, 2520, 903, 462, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,

%e 2992, 2860, 814, 66, 44, 44,

%e 3749, 2990, 1564, 644, 115, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,

%e ...

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

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

%K nonn,tabf

%O 3,2

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Jan 25 2020

%E Added "regular" to definition. - _N. J. A. Sloane_, Mar 06 2021