Sharings: distribution of distinguishable objects

We consider now, some mathematical models for distribution of objects. How to describe the distribution, for instance, of distinguishable objects into distinguishable boxes. Well, let us consider, for instance, a set of 7 numbers, and we want to distribute them into 4 level boxes, box number 1, box number 2, box number 3, books number 4. Well, we can put 1 into box number 3, 2 into box number 1, and so on.

What we obtain is what we call a 4-sharing of the set I7. More precisely, if we have two integers, k and n, we call n-sharing of Ik a sequence of n sets, C1, Cn, [INAUDIBLE] pairwise disjoint. Some of them may be empty, and these are subsets of Ik, whose union is the entire set Ik. So C1 is a subset of Ik that goes into box number 1, and Cn is the subset of Ik that goes into the box number n, if we think in terms of boxes. Let’s see how sharings can be described in terms of sequences.

Consider, for instance, the distribution of 10 labelled tickets of a lottery among 12 students, where we can take the set of tickets that go to student number 1, and let us call it C1, C2 the set of tickets that go to student number 2, and so on. C12 with a set of tickets that go to student number 12. Some of these sets may be empty. Actually, for sure, some of them will be empty. Some students will receive no tickets. These sets are disjoint, and their union is all the set of the 10 tickets of the lottery. So the ordered sequence, C1, C12, is exactly 12-sharing of I10. But we can describe this distribution also in terms of the sequence.

Instead of regrouping the tickets that go to the students, we can align the tickets and write below the name of the students that receive such a ticket. So let us put the tickets in a row, and let us now consider the sequence of the students that receive ticket number 1. Let us call it x1. So x1 will be 1 of the 12 students. x2 receives ticket number 2, and x10 receives ticket number 10. So x1, x10 are elements of the 12 students, and they form an ordered sequence of 10 terms. So we can describe this distribution with a 10-sequence of the 12 elements of I12.

So the same distribution can be described in terms of a 12-sharing of I10, but also in terms of a 10-sequence of I12. In general, this gives a 1-to-1 correspondence between the n-sharings of Ik and the k-sequences of In. More precisely, we can share the elements of Ik to n sets, C1, Cn, but equivalently, we can align the elements of Ik and assign the number to each we want to give, element number 1, element number 2, element number k. So what we obtain is a k-sequence of In. For instance, let us describe the following distribution of books. We have book number 2 and number 3 that go to student number 1.

Book number 1 that goes to student number 2, and the books from 4 to 7 that go to student number 3. Unfortunately, student number 4 receives no books. Well, let us translate this distribution in terms of sharings and then in terms of sequences. Let us first describe this distribution in terms of sharings. So we take the 7 books, and we build 4 different bags. The first is the bag that goes to student number 1. So if we want book 2 and 3 to be given to student number 1, the first box will be formed with the books 2, 3. The second bag, that goes to student number 2, will have just the book number 1.

The third bag, that goes to student number 3, will have the books 4, 5, 6, 7, and the bag number 4 will be empty. So what we obtain is exactly a 4-sharing of I7. Alternatively, we can align the 7 books and below, write the name of the students that receives each of the books. So here, book number 1 goes to student number 2, book number 2 goes to student number 1, and so on. What we obtain is a 7-sequence of the students 1, 2, 3, 4. Actually, there is no 4 in the sequence, because student number 4 receives no books.

Now, as a corollary of this correspondence between n-sharings of Ik and k-sequences of In– well, thanks to the bisection principal– we obtain that counting the number of n-sharings of Ik is exactly as counting the number of k-sequences of In.