The art (combinatorial) of sitting at the table

A convex hexagon can be divided into triangles, through diagonals that do not cut each other in 14 different ways, so the sequence of Catalan numbers, which we talked about last week, is:

1, 2, 5, 14, 42, 132...

El arte (combinatoria) de sentarse a la mesa

The guideline that follows this sequence is not easy to deduce, because its N-simo term is given by the formula Cn = (2n)!/(n + 1)! n!

So, the 3rd term of the succession is 6!/4! 3! = 6x5x4x3x2/4x3x2x3x2 = 5.

Catalan numbers often appear in combinatorial problems and/or help solve them, as we will see in future deliveries.

Around the table

And speaking of combinatory, our "Outstanding User" Francisco Montesinos proposed an interesting problem (see comment 2 of last week) of which I offer a very simplified version, as a first step for those who want to delve into the topic :

How many different male-female configurations can we find around a table?

With only one diner, there are only two possibilities: a man or a woman.

With two diners, there are 3 possibilities: Two men, two women, one man and one woman.

With three diners, there are four possibilities: three men, three women, two men and one woman, two women and one man.

With four diners there are 6 possibilities: Four men, four women, three men and one woman, three women and one man, two men and two women seated alternately (the well-known boy-girl-boy-girl protocol), two men and two women seated do not Alternately. (equal configurations are considered that can be superimposed by rotation; So this is the first case where a certain number of men and women give rise to two different configurations).

and five diners? What about six? What pattern follows the sequence?

Problems with people sitting around a table are a classic of recreational mathematics. Let's look at some simple examples as a reminder or introduction to the topic:

Three heterosexual couples: Antonio and Berta, Carlos and Diana, Ernesto and Fatima, sit around a round table. How many different ways can they do it? (as before, it is not considered different the superimposed configurations by rotation).

How many different ways can you sit so that Carlos and Diana, who are angry, are not together?

How many different ways can you sit respecting the alternation protocol (boy-girl-boy-girl...)?

How many different ways can the protocol alternation be skipped?

How many different ways can you sit with all the men on one side and all the women on the other?