We call a sum a1 + a2 + · · ·+ ak a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that a1 ≤ a2 ≤ · · · ≤ ak and n = a1 + a2 + · · ·+ ak. For i = 1, 2, . . . , k, we call ai the i-th part of the sum a1 + a2 + · · ·+ ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1 + a2 + · · · + ak and b1 + b2 + · · ·+ bk strongly inter...

We give a simple combinatorial proof of three identities of Warnaar. The proofs exploit involutions due to Franklin and Schur.

Theorems in the theory of partitions are closely related to basic hypergeometric series. Some identities arising in basic hypergeometric series can be interpreted in the theory of partitions using Fpartitions. In this paper, Ramanujan’s 1ψ1 summation and the q-Gauss summation are established combinatorially.

Single Nucleotide Polymorphisms (SNPs) are the most usual form of polymorphism in human genome.Analyses of genetic variations have revealed that individual genomes share common SNP-haplotypes. Theparticular pattern of these common variations forms a block-like structure on human genome. In this work,we develop a new method based on the Perfect Phylogeny Model to identify haplo...

this study investigated intra-specific variation in the growth parameters and dry mat-ter partitioning of sugar beet cultivars at different levels of salinity (0, 50, 150, 250 and 350 mm nacl + cacl2 in 5:1 molar ratio) using four cvs, one british (madison) and three iranian cvs (7233-p12, 7233-p21 and 7233-p29). the plants were grown in a controlled greenhouse environment for about 18 weeks. a...

background: sediments are inseparable fractions of wetland ecosystems that work as a his-torical archive in the process of recording heavy metal aggregation changes. this study was carried out for assessing co, ni, and zn in sediment samples of mighan wetland using geo-accumulation index in 2013. methods:the sediment samples were taken from 11 stations. the samples were subjected to bulk digest...