A367190 -id:A367190 - cp's OEIS frontend

A366253 Table read by antidiagonals: Place k points in general position on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives number of regions in the resulting planar graph.

1, 13, 4, 82, 67, 11, 307, 406, 206, 24, 841, 1441, 1216, 489, 50, 1891, 3796, 4211, 2835, 995, 80, 3718, 8299, 10901, 9672, 5671, 1802, 154, 6637, 15982, 23536, 24780, 19139, 10196, 3052, 220, 11017, 28081, 44906, 53109, 48686, 34166, 17011, 4810, 375

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Nov 09 2023

"In general position" implies that the internal lines (or chords) formed from the n*k edge points only have simple intersections; there is no interior points where three or more such chords meet. Note that for even-n n-gons, with n>=6, the chords from the n corner points do create non-simple intersections.

Note that although the number of regions with a given number of edges in the graph will vary as the edge points change position, the total number of regions will stay constant as long as all internal vertices created from the edge-point chords remain simple.

			The table begins:
1, 13, 82, 307, 841, 1891, 3718, 6637, 11017, 17281, 25906, 37423, 52417,...
4, 67, 406, 1441, 3796, 8299, 15982, 28081, 46036, 71491, 106294, 152497,...
11, 206, 1216, 4211, 10901, 23536, 44906, 78341, 127711, 197426, 292436,...
24, 489, 2835, 9672, 24780, 53109, 100779, 175080, 284472, 438585, 648219,...
50, 995, 5671, 19139, 48686, 103825, 196295, 340061, 551314, 848471, 1252175,...
80, 1802, 10196, 34166, 86480, 183770, 346532, 599126, 969776, 1490570,...
154, 3052, 17011, 56611, 142696, 302374, 569017, 982261, 1588006, 2438416,...
220, 4810, 26705, 88495, 222400, 470270, 883585, 1523455, 2460620, 3775450,...
375, 7305, 40096, 132243, 331431, 699535, 1312620, 2260941, 3648943, 5595261,...
444, 10509, 57810, 190263, 475980, 1003269, 1880634, 3236775, 5220588, 8001165,...
781, 14938, 81082, 265747, 663391, 1396396, 2615068, 4497637, 7250257,...
952, 20335, 110439, 361354, 900844, 1894347, 3544975, 6093514, 9818424,...
1456, 27391, 147421, 480931, 1197076, 2514781, 4702741, 8079421, 13013056,...
1696, 35716, 192552, 627484, 1560352, 3275556, 6122056, 10513372,...
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Scott R. Shannon, Image for T(5,3).
Scott R. Shannon, Image for T(6,2).
Scott R. Shannon, Image for T(8,2).
Scott R. Shannon, Image for T(10,2).

Cf. A367118 (first row), A367121 (second row), A007678 (first column), A367183 (vertices), A367190 (edges).

T(n,k) = A367190(n,k) - A367183(n,k) + 1 by Euler's formula.
Conjectured:
T(3,k) = A367118(k) = (9/4)*k^4 + 3*k^3 + (15/4)*k^2 + 3*k + 1.
T(4,k) = A367121(k) = (17/2)*k^4 + 19*k^3 + (43/2)*k^2 + 14*k + 4.
T(5,k) = (45/2)*k^4 + 60*k^3 + 70*k^2 + (85/2)*k + 11.
T(6,k) = (195/4)*k^4 + (285/2)*k^3 + (687/4)*k^2 + 102*k + 24.
T(7,k) = (371/4)*k^4 + 287*k^3 + (1421/4)*k^2 + 210*k + 50.
T(8,k) = 161*k^4 + 518*k^3 + 655*k^2 + 388*k + 80.
T(9,k) = 261*k^4 + 864*k^3 + (2223/2)*k^2 + (1323/2)*k + 154.
T(10,k) = (1605/4)*k^4 + (2715/2)*k^3 + (7085/4)*k^2 + 1060*k + 220.

A367183 Table read by antidiagonals: Place k points in general position on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives number of vertices in the resulting planar graph.

Original entry on oeis.org

3, 12, 5, 72, 58, 10, 282, 375, 185, 19, 795, 1376, 1155, 451, 42, 1818, 3685, 4090, 2734, 938, 57, 3612, 8130, 10700, 9478, 5523, 1711, 135, 6492, 15743, 23235, 24463, 18858, 9981, 2943, 171, 10827, 27760, 44485, 52639, 48230, 33771, 16740, 4646, 341

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Nov 08 2023

			The table begins:
3, 12, 72, 282, 795, 1818, 3612, 6492, 10827, 17040, 25608, 37062, 51987,...
5, 58, 375, 1376, 3685, 8130, 15743, 27760, 45621, 70970, 105655, 151728,...
10, 185, 1155, 4090, 10700, 23235, 44485, 77780, 126990, 196525, 291335,...
19, 451, 2734, 9478, 24463, 52639, 100126, 174214, 283363, 437203, 646534,...
42, 938, 5523, 18858, 48230, 103152, 195363, 338828, 549738, 846510, 1249787,...
57, 1711, 9981, 33771, 85849, 182847, 345261, 597451, 967641, 1487919, 2194237,...
135, 2943, 16740, 56106, 141885, 301185, 567378, 980100, 1585251, 2434995,...
171, 4646, 26336, 87831, 221351, 468746, 881496, 1520711, 2457131, 3771126,...
341, 7128, 39666, 131450, 330165, 697686, 1310078, 2257596, 3644685, 5589980,...
313, 10204, 57199, 189214, 474361, 1000948, 1877479, 3232654, 5215369, 7994716,...
728, 14677, 80457, 264602, 661570, 1393743, 2611427, 4492852, 7244172,...
771, 19909, 109586, 359892, 898591, 1891121, 3540594, 6087796, 9811187,...
1380, 27030, 146565, 479370, 1194600, 2511180, 4697805, 8072940, 13004820,...
1393, 35085, 191353, 625477, 1557297, 3271213, 6116185, 10505733,......
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Scott R. Shannon, Image for T(5,3).
Scott R. Shannon, Image for T(6,2).
Scott R. Shannon, Image for T(7,1).
Scott R. Shannon, Image for T(8,1).

Cf. A367117 (first row), A334698 (second row), A007569 (first column), A366253 (regions), A367190 (edges).

T(n,k) = A367190(n,k) - A366253(n,k) + 1 by Euler's formula.
T(3,k) = A367117(k) = (9/4)*k^4 + 3*k^3 + (3/4)*k^2 + 3*k + 3.
Conjectured:
T(4,k) = A334698(k+1) = (17/2)*k^4 + 19*k^3 + (31/2)*k^2 + 10*k + 5.
T(5,k) = (45/2)*k^4 + 60*k^3 + 60*k^2 + (65/2)*k + 10.
T(6,k) = (195/4)*k^4 + (285/2)*k^3 + (627/4)*k^2 + 84*k + 19.
T(7,k) = (371/4)*k^4 + 287*k^3 + (1337/4)*k^2 + 182*k + 42.
T(8,k) = 161*k^4 + 518*k^3 + 627*k^2 + 348*k + 57.
T(9,k) = 261*k^4 + 864*k^3 + (2151/2)*k^2 + (1215/2)*k + 135.
T(10,k) = (1605/4)*k^4 + (2715/2)*k^3 + (6905/4)*k^2 + 990*k + 171.

cp's OEIS Frontend

A366253 Table read by antidiagonals: Place k points in general position on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives number of regions in the resulting planar graph.

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