منابع مشابه
A note on partitions into distinct parts and odd parts
Bousquet-Mélou and Eriksson showed that the number of partitions of n into distinct parts whose alternating sum is k is equal to the number of partitions of n into k odd parts, which is a refinement of a well-known result by Euler. We give a different graphical interpretation of the bijection by Sylvester on partitions into distinct parts and partitions into odd parts, and show that the bijecti...
متن کاملThe Arithmetic of Partitions into Distinct Parts
so that Q(n) is odd if and only if n is a pentagonal number. This fact was generalized by Gordon and Ono [4], who demonstrated that for any positive integer k almost all values of Q(n) are divisible by 2k, and by Ono and Penniston [7], who found an exact formula for Q(n) modulo 8. Their techniques do not apply to odd primes, however, and for these primes the situation seems to be more difficult...
متن کاملPartitions of rt(t, n) into parts
Szekeres proved, using complex analysis, an asymptotic formula for the number of partitions of n into at most k parts. Canfield discovered a simplification of the formula, and proved it without complex analysis. We re-prove the formula, in the asymptotic regime when k is at least a constant times √ n, by showing that it is equivalent to a local central limit theorem in Fristedt’s model for rand...
متن کاملPartitions into Distinct Parts and Elliptic Curves
A positive integer D is called a ‘congruent number’ if there exists a right triangle with rational sidelengths with area D. Over the centuries there have been many investigations attempting to classify the congruent numbers, but little was known until Tunnell [T] brilliantly applied a tour de force of methods and provided a conditional solution to this problem. It turns out that a square-free i...
متن کاملMaximum Product Over Partitions Into Distinct Parts
We establish an explicit formula for the maximum value of the product of parts for partitions of a positive integer into distinct parts (sequence A034893 in the On-Line Encyclopedia of Integer Sequences).
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ژورنال
عنوان ژورنال: Mathematics Magazine
سال: 2005
ISSN: 0025-570X
DOI: 10.2307/30044200