1,2

Number of chambers in an n-sected triangle. That is, n sectors are extended from each vertex to the opposite edge of the triangle. - Eric Gottlieb (gottlieb(AT)rhodes.edu), Jun 26 2005

How many chambers does the edge n-sected simplex with m vertices have? We have given just the first few terms of the case m = 3. This question is natural in the context of central hyperplane arrangements as it generalizes the braid arrangement. Mike Ackerman, Sul-Young Choi, Peter Coughlin, Japheth Wood and I originally encountered the question in the context of voting theory, where we were exploring ways to tabulate votes when voters' preferences are partially ordered. Unfortunately, it turns out that the chambers of the 3-sected simplex with n vertices are not in correspondence with the set of posets on n letters as the chain with three elements and a fourth incomparable element illustrates. - Eric Gottlieb (gottlieb(AT)rhodes.edu), Jun 26 2005

"Equilateral" is not needed: the sequence counts regions correctly for any triangle with n-sected sides. Ceva's Theorem is used to deduct vanishing regions from the naive count. The first deduction is at n=15 for n odd, and n=20 for n even. - Len Smiley and Brian Wick (mathclub(AT)math.uaa.alaska.edu), Jul 04 2005

Hugo Pfoertner, Visualization of diagonal intersections in an equilateral triangle.

Sequences formed by drawing all diagonals in regular polygon

Note that 3 divides a(2k) and a(2k+1)-1. (T. D. Noe)

E.g. the number of chambers in the bisected triangle is six, the number of permutations on 3 letters. The number of chambers in the trisected triangle is equal to 19, the number of posets on 3 elements. - Eric Gottlieb (gottlieb(AT)rhodes.edu), Jun 26 2005

a(2)=6: The 3 line segments cut the equilateral triangle into 6 triangles.

a(3)=19: The 3*2 line segments form 12 triangles, 3 quadrilaterals, 3 pentagons and 1 central non-regular hexagon. See pictures at Pfoertner link.

regions:=proc(n::nonnegint) local j, k, l, a; a:=0; if (n mod 2<>0) then a:=3*n^2-3*n+1 else a:=3*n^2-6*n+6 fi; for l from 1 to floor(n/2)-1 do for k from 1 to floor(n/2)-1 do for j from 1 to floor(n/2)-1 do if((n-k)*l*j=k*(n-l)*(n-j)) then a:=a-6 fi od od od; return a end proc; seq(regions(i), i=1..100); (Len Smiley and Brian Wick)

(PARI) for(n=1, 100, regions=0; if(n%2!=0, regions=3*n^2-3*n+1, regions=3*n^2-6*n+6); for(l=1, floor(n/2)-1, for(k=1, floor(n/2)-1, for(j=1, floor(n/2)-1, if((n-k)*l*j==k*(n-l)*(n-j), regions-=6)))); print1(regions, ", ")) - Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 27 2006

Cf. A091908 number of intersections, A091910 radial locations of intersection points, A006533.

Sequence in context: A042215 A041911 A038125 this_sequence A063233 A063147 A031014

Adjacent sequences: A092095 A092096 A092097 this_sequence A092099 A092100 A092101

nonn

Hugo Pfoertner (hugo(AT)pfoertner.org), Feb 19 2004

More terms from T. D. Noe (noe(AT)sspectra.com), Jun 29 2005

Further terms from Brian Wick (mathclub(AT)math.uaa.alaska.edu), Jun 30 2005

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 27 2006