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If 2n people are sitting at equally spaced intervals around a circle. How many ways can they form n pairs if no two persons seated directly opposite each other can form a pair?

What I did was draw some examples for low values of 'n' and also tried to see it as a graph on 2n vertices where there is no edge between opposite vertices. However, I have not come to a revelation. Any help would be appreciated.

HumbleStudent
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2 Answers2

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Hint

$2n$ persons can form $\binom{2n}{2}$ pairs without restrictions.

How many pairs must we subtract from that due to the restriction ?

I am assuming that the $2n$ persons have already been seated in some order.

Then each of $n$ persons will have a specific person opposite with whom they can't form a pair,

thus $\binom{2n}{2} - n$ allowable pairs.

true blue anil
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As shown at the beginning of this answer, without restriction, there are $(2n-1)!!$ ways to form pairs with $2n$ people. Let us count how many ways there are to pair at least one opposite using Inclusion-Exclusion.

There are $\binom{n}{k}$ ways to choose $k$ opposites. For each of those choices, there are $(2n-2k-1)!!$ ways to form pairs with the $2n-2k$ left. Inclusion-Exclusion says that there are $$ \sum_{k=1}^n(-1)^{k-1}\binom{n}{k}(2n-2k-1)!! $$ pairings with at least one pair of opposites. Subtracting this from $(2n-1)!!$ gives $$ \bbox[5px,border:2px solid #C0A000]{\sum_{k=0}^n(-1)^k\binom{n}{k}(2n-2k-1)!!} $$ ways to pair without matching opposites. $$ \begin{array}{c|c} n&1&2&3&4&5&6&7&8&9\\\hline \text{pairings}&0&2&8&60&544&6040&79008&1190672&20314880 \end{array} $$ Here are the $8$ pairings for $n=3$:

enter image description here

Community
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robjohn
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  • What is your interpretation of the question ? Initially I though derangements, but "$2n$ people are sitting in a circle.." would seem to indicate that we need to work out pairings for any particular seating – true blue anil Dec 16 '15 at 07:15
  • @trueblueanil: I have added an image for $n=3$ which gives $8$ pairings. – robjohn Dec 16 '15 at 07:35
  • I understand your interpretation, you are simultaneously pairing all the people sitting in a circle, while I interpreted it as the sides and chords of a polygon excluding chords that pass through the center. – true blue anil Dec 16 '15 at 08:05
  • The question asks "How many ways can they form $n$ pairs if no two persons seated directly opposite each other can form a pair?" Each of the $8$ arrangements in the $n=3$ image above is "a way to form $3$ pairs if no two persons seated directly opposite each other can form a pair." – robjohn Dec 16 '15 at 08:12
  • I just noticed that the Wikipedia page for Double Factorial has an image for all pairings of $6$ points. – robjohn Dec 16 '15 at 13:10
  • Your answer is right for your interpretation. (+1) See https://oeis.org/A053871/internal – true blue anil Dec 16 '15 at 13:53
  • @trueblueanil: thanks for the link. That offers some interesting alternate interpretations of the same sequence. – robjohn Dec 16 '15 at 14:28
  • Regarding my interpretation: the question asks "how many ways can they form $n$ pairs", not "how many pairs can be formed". – robjohn Dec 16 '15 at 15:39
  • I agree with your interpretation, yeah OEIS always gives so many ideas ! – true blue anil Dec 16 '15 at 16:05