Timeline for Combinatorics related plane geometry
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2022 at 12:45 | vote | accept | Janaka Rodrigo | ||
| Mar 28, 2022 at 17:33 | comment | added | Janaka Rodrigo | Total number of different ways of partitioning the set of vertices of a convex n-gon into nonintersecting polygons is same as what is described in OEIS A 114997 as number of ordered trees with n edges and no unary or binary nodes. But I wanted to find different ways separately when partitioning into 1,2,3,... nonintersecting polygons and I did it upto partitioning into 5 polygons. Thereafter I used the pattern to derive general formula to cover all cases. | |
| Mar 28, 2022 at 1:46 | comment | added | Janaka Rodrigo | Since speed of the motions not mentioned and no one can meet another one you need to consider nonintersecting partitions. | |
| Mar 28, 2022 at 1:21 | comment | added | Richard Stanley | For the case where a man can stand still, we are looking at $$ \left( \frac{x(1-x)}{1-x^2+2x^3}\right)^{\langle -1\rangle} $$ $$ = x(1+x+x^2+3x^3+11x^4+33x^5+97x^6+311x^7+1047x^8+\cdots) $$ not (yet) in OEIS. | |
| Mar 28, 2022 at 1:15 | comment | added | Richard Stanley | The intended interpretation is that the paths don't cross, as confirmed by OEIS A350599. | |
| Mar 28, 2022 at 1:06 | comment | added | Gerry Myerson | If men are not allowed to stay put, it's oeis.org/A038205 derangements with cycle lengths at least three. | |
| Mar 28, 2022 at 0:59 | comment | added | Gerry Myerson | The way I read the question, it doesn't say the paths can't cross, it just says no two can be at the same place at the same time. That forces it to be a permutation, and rules out transpositions, but as long as they're careful with their timing it doesn't rule out crossing paths. | |
| Mar 27, 2022 at 23:29 | comment | added | Richard Stanley | @GerryMyerson: where does the noncrossing condition come in? | |
| Mar 27, 2022 at 22:20 | comment | added | Gerry Myerson | R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7 is cited at oeis.org/A000266 which counts "the number of permutations in the symmetric group S_n whose cycle decomposition contains no transposition." This should give the number of ways if men are allowed to stand still. | |
| Mar 27, 2022 at 2:58 | comment | added | Janaka Rodrigo | Yes as you suggested this is regarding how to form nonintersecting closed loops that means how many different ways are there to partition the set of vertices of a convex n - gon into nonintersecting directed polygons. I tried this as a research and the result published in OEIS A 350599 | |
| Mar 27, 2022 at 0:08 | history | answered | Richard Stanley | CC BY-SA 4.0 |