Partitions and polyominoes | The game of science

The seemingly impossible partition of an obtuse triangle into acu-angles, which we have dealt with for the last two weeks, requires that at least one common vertex of the acu-angles is not perimeter, but interior (see figure from the previous installment), and in that vertex must converge at least 5 angles, so that all are acute; therefore, the division requires a minimum of 7 acute angles: 5 interior and 2 more on both sides of the pentagon that they form.

As for the succession of the respective numbers of partitions of the natural numbers (and if it is considered that 0 has, like 1, a partition), it is the following:

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792…

The number of partitions of the natural number n is designated as p (n), and it is easy to see that its growth “accelerates” as n grows; So:

p(10) = 42

p(100) = 190.569.292

p(200) = 3.972.998.029.388

p(1000) = 24.061.467.864.032.622.473.692.149.727.991

There is no simple formula that gives us the number of partitions as a function of n; The famous mathematicians GH Hardy and Srinivasa Ramanujan studied the function p (n) at the beginning of the last century and came up with an interesting asymptotic expression (when n tends to infinity), too complex to deal with here.

My astute readers will not have missed the parallelism (identity, rather) between Young’s schemes for the representation of partitions, seen last week, and the polyominoes, to which we dedicate several installments of The game of science last year (see Polyominos, June 5, 2020, and earlier).

According to Young’s schemes, the partition 10 = 5 + 4 + 1 is represented as shown in the figure (the order of the addends is irrelevant, but they are usually arranged from highest to lowest):

This polyomino of order 10 is one of the 4,460 possible decomines (without holes), much more than the 42 partitions of 10 in addends: less than one hundredth part of the decomines are Young’s schemes. Can we draw any conclusions from this proportion or make any generalizations about it?

Ramanujan’s number

When discussing Hardy and Ramanujan in relation to the partitions of whole numbers into addends, it is inevitable to recall the famous anecdote concerning the number 1729, which was renamed the Ramanujan number. On one occasion, Hardy commented to the brilliant Indian mathematician that he had taken a taxi with the license number 1729 and found that number very uninteresting, to which Ramanujan replied: “Don’t say that, Hardy, 1729 is the least number that can be expressed in two different ways as the sum of two cubes ”.

Indeed, 1729 = 10³ + 9³ = 12³ + 1³; but is it really the smallest number with this property? And, by the way, does it make sense to talk about “uninteresting” numbers?

On the other hand, and without pretending to detract from the great Ramanujan, his feat of mental calculation, being undoubtedly remarkable, is not as amazing as it seems at first glance. Someone very familiar with numbers knows the cubes of the first integers well and it is not difficult for them to realize at first glance that 1728 = 12³ and 729 = 9³, and from there to see that adding 1 to the first cube and 1000 to the second you get 1729 there is only one step.

Carlo Frabetti is a writer and mathematician, member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and young people, including ‘Damn physics’, ‘Damn mathematics’ or ‘The great game’. He was a screenwriter for ‘La bola de cristal’.

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George Holan

George Holan is chief editor at Plainsmen Post and has articles published in many notable publications in the last decade.

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