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October 11

polygon disjoint sets

Let a polygon with m vertices labeled as 1,2,3,..,m. How many subsets of {1,2,..m} having two or more elements exists with the property that they should not include any two consecutive vertices of polygon. — Preceding unsigned comment added by 182.187.95.175 (talk) 21:29, 11 October 2014 (UTC)[reply]

Let's count this for subsets of size k, then sum over k. We'll break these into two categories: those containing 1, and those omitting it.

Let's start with those containing one. Then there are k "gaps"--one after every element. Each of these gaps has at least 1 element in it, which leaves (m-2k) elements unaccounted for. These remaining elements need to be distributed amongst the gaps. Stars and bars is good for counting the number of ways to do this, giving us

{\frac {(m-k-1)!}{(m-2k)!(k-1)!}}

different sets with k elements including 1.

Next, we'll consider those sets omitting 1. Here there are k+1 gaps--one after every element, and one before 1. Each has at least 1 element, except possibly the last one. So still (m-2k) elements unaccounted for, but this time k+1 many gaps to distribute them in. So

{\frac {(m-k)!}{(m-2k)!k!}}

sets with k elements and excluding 1.

Note that k can be at most half of m, so the total number of sets is

\sum _{k=2}^{m/2}{\frac {(m-k-1)!}{(m-2k)!(k-1)!}}+{\frac {(m-k)!}{(m-2k)!k!}}

. The interior can be simplified a bit, getting you

\sum _{k=2}^{m/2}{\frac {m(m-k-1)!}{(m-2k)!k!}}

.--80.109.80.31 (talk) 23:19, 11 October 2014 (UTC)[reply]

This is A023548. I don't know why it is, but the first 20 terms all match. -- BenRG (talk) 00:42, 12 October 2014 (UTC)[reply]

As perhaps a clue, I understand why the terms representing the prime m have to be divisible by m in the polygon disjoint sets, if we understood why it was true in A023548... (note A023548 needs to be shifted by 3, since a square is the first one that can be disjoint...Naraht (talk) 01:05, 12 October 2014 (UTC)[reply]

It's well known that the Lucas numbers represent the number of cyclic 0-1 sequences that have no consecutive 1's; it's not mentioned in our article but the oeis entry OEIS: A000204 mentions it. It's a small leap to get from that the numbers of subsets of the vertices of a polygon with no consecutive elements are Lucas numbers. You can also state this as the number of independent sets in a cyclic graph. The number of sets with 0 or 1 element is easy to determine so that gives a formula for the number in question. Lucas numbers can be given in terms of Fibonacci numbers so it's probably not hard to turn this formula into a proof that the sequence in question is A023548. --RDBury (talk) 13:21, 12 October 2014 (UTC)[reply]