Coins in a fountain
In combinatorial mathematics, coins in a fountain is an interesting problem with an even more interesting generating function. The problem is described below:
In how many different number of ways can you arrange n coins with k coins at the base such that all the coins above the base layer touch exactly two coins of the lower layer.[1]
Solution:
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1 | 1 | 2 | 3 | 5 | 9 | 15 | 26 | 45 | 78 | 135 | 234 | ... |
The above sequence show the number of ways in which n coins can be stacked. So, for example for 9 coins we have 45 different ways of stacking them in a fountain. The number which is the solution for the above stated problem is then given by the coefficients of the polynomial of the following generating function:
(1) |
Such generating function are extensively studied in[4]
Specifically, the number of such fountains that can be created using n coins is given by the coefficients of:
(2) |
This is easily seen by substituting the value of y to be 1. This is because, suppose the generating function for (1) is of the form:
then, if we want to get the total number of fountains we need to do summation over k. So, the number of fountains with n total coins can be given by:
which can be obtained by substituting the value of y to be 1 and observing the coefficient of xn.
Proof of generating function (1). Consider the number of ways of forming a fountain of n coins with k coins at base to be given by . Now, consider the number of ways of forming the same but with the restriction that the second most bottom layer (above the base layer) contains no gaps, i.e. it contains exactly k − 1 coins. Let this be called primitive fountain and denote it by . The two functions are related by the following equation:
(3) |
This is because, we can view the primitive fountain as a normal fountain of n − k' coins with k − 1 coins in the base layer staked on top of a single layer of k coins without any gaps. Also, consider a normal fountain with a supposed gap in the second last layer (w.r.t. the base layer) in the r position. So, the normal fountain can be viewed as a set of two fountains:
- A primitive fountain with n' coins in it and base layer having r coins.
- A normal fountain with n − n' coins in it and the base layer having k − r coins.
So, we get the following relation:
(4) |
Now, we can easily observe the generating function relation for (4) to be:
(5) |
and for (3) to be:
(6) |
Now, simply substituting (6) in (5) we get the relation:
References
- ^ Odlyzko, A. M. and Wilf, H. S. (1988) The editor’s corner: n coins in a fountain. Amer. Math. Monthly 95 840–843.[1]
- ^ Sloane, N. J. A. (2000) The On-Line Encyclopedia of Integer Sequences. Published electronically at "Sloane's encyclopedia of sequences"
- ^ Phillipe Duchon, Phillipe Flajolet, Guy Louchard and Gilles Schaeffer (2003)"Boltzmann Samplers for the Random Generation of Combinatorial Structures"
- ^ Flajolet, P. (1980) Combinatorial aspects of continued fractions. Discrete Math. 32 125–161.