Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
"Keith F. Lynch" <kfl@KeithLynch.net> wrote:
What's needed is a way to look at every possible arrangement of n points. By "arrangement" I don't mean the uncountable infinity of possible locations of points in the unit square, I mean a difference that makes a difference.
How would one go about enumerating all arrangements?
This looks relevant: http://www.ist.tugraz.at/staff/aichholzer/research/rp/triangulations/orderty...
Thanks. But that apparently just points to abstracts of papers that aren't online, and binary data files in undocumented formats. I re-ran my program on two billion arrangements of seven points, and counted the RCNs of each. This time, I discarded any arrangements that contained any vertical line. Earlier I had just not worried about the resulting divide-by-zero errors, as those would cause errors only in the ninth digit or later. I also checked for the other possible source of division by zero, parallel lines, but as always those never happened. I did not check for three points in a line. Anyhow, I got surprising results. There are no longer just 13 possible RCNs for n=7, there are 24: 9: 2854081 10: 1 11: 11027816 12: 1 13: 23633438 14: 6 15: 57311057 16: 2 17: 146814179 18: 4 19: 239174223 20: 6 21: 275889258 22: 4 23: 384422525 24: 1 25: 309111456 26: 2 27: 205481998 28: 2 29: 180328085 30: 1 31: 116777008 35: 47174846 I'm not sure this isn't a bug. Can someone come up with a proof that there can't be any even-numbered RCN for 7 points? Or that there can be? I got similar results for n=9, except that this time it's the even numbers that are very common and the odd numbers that are very rare: 36: 13927 38: 66278 40: 180469 42: 327642 44: 533071 46: 890830 47: 1 48: 1463260 49: 1 50: 2562426 51: 7 52: 4240140 53: 8 54: 6419703 55: 2 56: 10012353 57: 12 58: 15368762 59: 22 60: 21885908 61: 25 62: 31885941 63: 50 64: 44747612 65: 28 66: 57233553 67: 74 68: 75238774 69: 76 70: 93957198 71: 67 72: 105470193 73: 86 74: 121665340 75: 78 76: 132425970 77: 73 78: 133795238 79: 69 80: 142405957 81: 61 82: 139308013 83: 64 84: 126803508 85: 45 86: 125471727 87: 32 88: 110246476 89: 25 90: 93769089 91: 23 92: 89460437 93: 20 94: 67815191 95: 7 96: 54064073 97: 19 98: 54720289 99: 5 100: 33644869 101: 3 102: 26704075 103: 6 104: 26087965 105: 4 106: 11087442 107: 1 108: 13027353 110: 11485129 112: 1437325 113: 1 114: 5655861 116: 3348265 120: 2333611 126: 737762 I haven't done the full run for higher n yet, but there are preliminary signs that n=11 and n=17 will be almost but not quite all evens, and that n=13 and n=15 will be almost but not quite all odds. There's no sign of these biases for n=even.