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A weak composition of an integer s with m parts is a way of writing s as the sum of a sequence of non-negative integers of length m. Given two positive integers m and n, let N(m, n) denote the set of all weak compositions α = (α 1 , ⋯ , α m) with 0 ≤ α i ≤ n for 1 ≤ i ≤ m and c w m , n (s) be the number of weak composition of s into m parts with no part exceeding n. A poset is called a symmetric chain decomposition if the poset can be expressed as a disjoint union of symmetric chains. In this paper, we show that the poset N(m, n) can be expressed as a disjoint of symmetric chains by constructive method, which implies that its rank sequence c w m , n (0) , c w m , n (1) , ⋯ , c w m , n (m n) is unimodal and symmetric. [ABSTRACT FROM AUTHOR]

In chemistry, Cyvin-Gutman enumerates Kekulé numbers for certain benzenoids and record it as A 050446 on OEIS. This number is exactly the two variable array T (n , m) defined by the recursion T (n , m) = T (n , m − 1) + ∑ k = 0 ⌊ n − 1 2 ⌋ T (2 k , m − 1) T (n − 1 − 2 k , m) , where T (n , 0) = T (0 , m) = 1 for all nonnegative integers m , n. Interestingly, this number also appeared in the context of weighted graphs, graph polytopes, magic labellings, and unit primitive matrices, studied by different authors. Several interesting conjectures were made on the OEIS. These conjectures are related to both the row and column generating function of T (n , m). In this paper, we give explicit formula of the column generating function, which is also the generating function F (n , x) studied by Bóna, Ju, and Yoshida. We also get trig function representations by using Chebyshev polynomials of the second kind. This allows us to prove all these conjectures. [ABSTRACT FROM AUTHOR]

• We give a combinatorial proof of the parity unimodality of the (3, n) rational Catalan polynomials Q 3 , n (q). The idea is that we construct a poset whose rank generating function is Q 3 , n (q). Then we find that this poset has an identical domino chain decomposition, which implies the parity unimodality of Q 3 , n (q). A sequence a 0 , a 1 , a 2 , ⋯ is parity unimodal if it exists k 1 and k 2 such that a 0 ≤ a 2 ≤ ⋯ ≤ a k 1 ≥ a k 1 + 2 ≥ ⋯ and a 1 ≤ a 3 ≤ ⋯ ≤ a k 2 ≥ a k 2 + 2 ≥ ⋯. A polynomial f (q) = ∑ i = 0 n a i q i is parity unimodal if its coefficient sequence a 0 , a 1 , ⋯ , a n is parity unimodal. Recently, Xin and Zhong conjectured that the (m , n) -rational q -Catalan polynomial is parity unimodal. They showed that this conjecture holds for m ≤ 5 using the constant term method and generating functions. In this paper, we give a combinatorial proof of the m = 3 case. [ABSTRACT FROM AUTHOR]

Molecular detection methods are often used to detect the presence of common pathogenic bacteria. However, molecular detection methods for Yersinia enterocolitica (Y. enterocolitica) , an important foodborne pathogen, are rarely reported. This is attributable to the need for better detection targets. Therefore, this study aims to obtain a species-specific target for detecting Y. enterocolitica through pan-genome analysis using public genome resources. The YE5303_14291 gene was identified as a candidate target, and a YE5303_14291 -based CRISPR/Cas12a fluorescence visualization and test strip platform were developed. The developed PCR-CRISPR/Cas12a and RAA-CRISPR/Cas12a systems can quantitatively detect Y. enterocolitica , with linear ranges from 3.1 × 1010 CFU mL−1 to 3.1 × 10° CFU mL−1, and the limit of detection (LOD) was 3.1 × 10° CFU mL−1. The detection systems all showed reasonable specificity, and the actual sample detection results were consistent with the isolation and culture results. In conclusion, the CRISPR/Cas12a detection system based on the target YE5303_ 14291 has the potential to specifically and rapidly detect Y. enterocolitica in food samples. • Mining of species-specific targets of Y. enterocolitica using pan-genomic analysis. • Compared with previous articles, the target YE5303_ 14291 has better specificity. • The CRISPR/Cas12a system based on YE5303_14291 can detect Y. enterocolitica in pork. [ABSTRACT FROM AUTHOR]

A magic labelling of a graph G with magic sum s is a labelling of the edges of G by nonnegative integers such that for each vertex v ∈ V , the sum of labels of all edges incident to v is equal to the same number s. Stanley gave remarkable results on magic labellings, but the distinct labelling case is much more complicated. We consider the complete construction of all magic labellings of a given graph G. The idea is illustrated in detail by dealing with three regular graphs and a non-regular graph. We give combinatorial proofs. The structure result was used to enumerate the corresponding magic distinct labellings. The idea can be carried out directly for graphs with no more than 9 edges. [ABSTRACT FROM AUTHOR]

Yersinia enterocolitica (Y. enterocolitica) is an important foodborne pathogen that causes gastrointestinal syndromes. The objectives of this study were to examine the incidence of Y. enterocolitica strains in food samples in China and determine their genotypes, in addition to assessing their antimicrobial resistance and pathogenic potential. From June 2015 to June 2016, a total of 1588 food samples were obtained, and 37 (2.33%) were found to be contaminated with Y. enterocolitica. A total of 51 different Y. enterocolitica strains were isolated. All isolates belonged to biotype 1A, and the predominant serotype was O:5. All strains lacked the ail , virF , ystA , and yadA virulence genes, but most strains (>75%) harbored the ymoA , sat , hreP , fepD , and fes genes. Over 85% of the 51 isolated strains were sensitive to tetracycline, trimethoprim/sulfamethoxazole, and chloramphenicol, whereas they were resistant to ampicillin and amoxicillin/clavulanic acid. Furthermore, all the strains were divided into seven clusters according to the results of the enterobacterial repetitive intergenic consensus-polymerase chain reaction. Our research emphasizes, for the first time, that more stringent cold storage methods may increase the risk of Y. enterocolitica contamination. • Y. enterocolitica with the bioserotype 1A/O:5 dominates in retail food in China. • The contamination rate in supermarket is higher than that in traditional market. • The drug resistance of foodborne Y. enterocolitica has regional characteristics. [ABSTRACT FROM AUTHOR]