Construction of Polite Representations From Odd Divisors
To see the connection between odd divisors and polite representations, suppose a number x has the odd divisor y > 1. Then y consecutive integers centered on x/y (so that their average value is x/y) have x as their sum:
Some of the terms in this sum may be zero or negative. However, if a term is zero it can be omitted and any negative terms may be used to cancel positive ones, leading to a polite representation for x. (The requirement that y > 1 corresponds to the requirement that a polite representation have more than one term; applying the same construction for y = 1 would just lead to the trivial one-term representation x = x.) For instance, the polite number x = 14 has a single nontrivial odd divisor, 7. It is therefore the sum of 7 consecutive numbers centered at 14/7 = 2:
- 14 = (2 − 3) + (2 − 2) + (2 − 1) + 2 + (2 + 1) + (2 + 2) + (2 + 3).
The first term, −1, cancels a later +1, and the second term, zero, can be omitted, leading to the polite representation
- 14 = 2 + (2 + 1) + (2 + 2) + (2 + 3) = 2 + 3 + 4 + 5.
Conversely, every polite representation of x can be formed from this construction. If a representation has an odd number of terms, x/y is the middle term, while if it has an even number of terms and its minimum value is m it may be extended in a unique way to a longer sequence with the same sum and an odd number of terms, by including the 2m − 1 numbers −(m − 1), −(m − 2), ..., −1, 0, 1, ..., −(m − 2), −(m − 1). After this extension, again, x/y is the middle term. By this construction, the polite representations of a number and its odd divisors greater than one may be placed into a one-to-one correspondence, giving a bijective proof of the characterization of polite numbers and politeness. More generally, the same idea gives a two-to-one correspondence between, on the one hand, representations as a sum of consecutive integers (allowing zero, negative numbers, and single-term representations) and on the other hand odd divisors (including 1).
Another generalization of this result states that, for any n, the number of partitions of n into odd numbers having k distinct values equals the number of partitions of n into distinct numbers having k maximal runs of consecutive numbers. Here a run is one or more consecutive values such that the next larger and the next smaller consecutive values are not part of the partition; for instance the partition 10 = 1 + 4 + 5 has two runs, 1 and 4 + 5. A polite representation has a single run, and a partition with one value d is equivalent to a factorization of n as the product d×(n/d), so the special case k = 1 of this result states again the equivalence between polite representations and odd factors (including in this case the trivial representation n = n and the trivial odd factor 1).
Read more about this topic: Polite Number
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Sir Nathaniel. A most singular and choice epithet.”
—William Shakespeare (1564–1616)