April 20, 2021

Bell numbers | Science

Bell numbers | Science



As we saw, our three traveling friends last week They could go in one, two or three cars in five different ways. If there was one more friend, they could each go in their car (ABCD), the four together (ABCD), two together and the other two separated in six different ways (AB-CD, AC-BD, AD-BC, BC- AD, BD-AC, CD-AB), two in one car and two in another in three different ways (AB-CD, AC-BD, AD-BC) or one in one car and three in another in four different ways ( A-BCD, B-ACD, C-ABD, D-ABC), in total, 1 + 1 + 6 + 3 + 4 = 15 possibilities.

If the friends are five, the possibilities are 52, and if there are six, 203. The successive terms grow rapidly, and if we include the cases of a single traveler (1 possibility) and two travelers (2 possibilities: both in the same car or each one in yours), the sequence is as follows: 1, 2, 5, 15, 52, 203, 877, 4140, 21147 …

Each term of this sequence is the number of partitions of a set of n elements, with partitions being the different ways in which the set can be divided into complementary subsets, that is, without common elements and whose union is the complete set (as in the case of traveling friends). A set of one element only supports one partition. A set of two elements supports two different partitions: the two together or each one separately. A set of three elements supports five partitions, as we have seen in the case of the three friends, and so on.

The number of partitions in a set is called Bell's number, in honor of the Scottish mathematician and writer Eric Temple Bell, whom science fiction fans may know more by his pseudonym John Taine, author of classics of the genre such as The purple sapphire Y Seeds of life. The Bell number is usually expressed by the letter B with a subscript indicating the number of elements of the corresponding set; well, Bone= 1, Btwo= 2, B3= 5 … The succession of the Bell numbers usually starts with two ones: 1, 1, 2, 5, 15 …, because the case of the empty set is also considered, B0= 1

Factorizations

A composite number, by definition, can be broken down into primes and, therefore, expressed as a product of several whole numbers; if it only has two divisors (other than itself and the unit), this expression is unique; for example, 21 = 3 x 7 (in addition to the trivial 1 x 21). But if a number has more than two divisors, it can be expressed as a product of other numbers in different ways, in addition to the usual decomposition into prime factors; for example, 66 = 2 x 3 x 11 = 6 x 11 = 2 x 33 …

In how many different ways can the number 210 be expressed as a product of several numbers (integers, it is understood)? And the 2.310? Do these factorizations have anything to do with what has been seen before?

I invite my astute readers to examine the interesting and versatile succession of Bell's numbers and to share their conclusions.

Carlo Frabetti He is a writer and mathematician, a member of the New York Academy of Sciences. He has published more than 50 scientific dissemination works for adults, children and young people, among them Damn physics, Damn mathematics or The big game. He was a screenwriter The Cristal ball.

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