MathDB

Christian Reiher and Reid Barton want to open a security box, they already managed to discover the algorithm to generate the key codes and they obtained the following information:

i)

In the screen of the box will appear a sequence of

n+1

numbers,

C_0 = (a_{0,1},a_{0,2},...,a_{0,n+1})

ii)

If the code

K = (k_1,k_2,...,k_n)

opens the security box then the following must happen:
a) A sequence

C_i = (a_{i,1},a_{i,2},...,a_{i,n+1})

will be asigned to each

k_i

defined as follows:

a_{i,1} = 1

and

a_{i,j} = a_{i-1,j}-k_ia_{i,j-1}

, for

i,j \ge 1

b) The sequence

(C_n)

asigned to

k_n

satisfies that

S_n = \sum_{i=1}^{n+1}|a_i|

has its least possible value, considering all possible sequences

K

.
The sequence

C_0

that appears in the screen is the following:

a_{0,1} = 1

and

a_0,i

is the sum of the products of the elements of each of the subsets with

i-1

elements of the set

A =

{

1,2,3,...,n

i\ge 2

, such that

a_{0, n+1} = n!

Find a sequence

K = (k_1,k_2,...,k_n)

that satisfies the conditions of the problem and show that there exists at least

n!

of them.

Problem Statement