منابع مشابه
Identities Relating the Number of Partitions into an Even and Odd Number of Parts
If / > 0 and n > 1, let qf (n) denote the number of partitions of/? into an even number of parts, where each part occurs at most / times and Set qf(n) denote the number of partitions of n into an odd number of parts, where each part occurs at most/times. \ii>0, \etqf(0)= 1 andqf(0) = 0. For/ >Oandn > 0, \v\Aj(n) = qJ(n)-q°(n). For/= 1, it is well known [1] that , A (n) = I ~ i f n = f ^ orsome ...
متن کاملCounting Even and Odd Partitions
1. Introduction. It is a lovely fact that n] = f1; 2; : : : ; ng, n 1, has as many subsets X with an even cardinality jXj as those with an odd cardinality, namely 2 n?1 of both. To prove it, pair every subset X with X 1 where X 1 is Xnf1g if 1 2 X and X f1g if 1 6 2 X. Then X 7 ! X 1 is an involution that changes the parity of jXj and the result follows. More generally, in enumerative combinato...
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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 کاملAsymptotic Behavior of Odd-Even Partitions
Andrews studied a function which appears in Ramanujan’s identities. In Ramanujan’s “Lost” Notebook, there are several formulas involving this function, but they are not as simple as the identities with other similar shape of functions. Nonetheless, Andrews found out that this function possesses combinatorial information, odd-even partition. In this paper, we provide the asymptotic formula for t...
متن کاملSynchrotron Radiation Studies on Even-Odd and Odd-Even Nylons
Aliphatic polyamides derived from odd diamine or odd dicarboxylic acid units cannot adopt a conventional sheet structure when molecular chains have an all trans conformation. However, typical fiber diffraction patterns of this sheet structure were observed in several polyamides derived from odd units such as nylons 65 and 56. Consequently, a new structure based on the establishment of intermole...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2016
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2016.02.031