plane trees
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2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Ankit Bisain ◽  
Eric Hanson

Bernardi has given a general formula for the number of regions of a deformation of the braid arrangement as a signed sum over boxed trees. We prove that each set of boxed trees which share an underlying (rooted labeled plane) tree contributes 0 or $\pm 1$ to this sum, and we give an algorithm for computing this value. For Ish-type arrangements, we further construct a sign-reversing involution which reduces Bernardi's signed sum to the enumeration of a set of (rooted labeled plane) trees. We conclude by explicitly enumerating the trees corresponding to the regions of Ish-type arrangements which are nested, recovering their known counting formula.


Molecules ◽  
2021 ◽  
Vol 26 (18) ◽  
pp. 5583
Author(s):  
Hanghang Lou ◽  
Hao Li ◽  
Shengliang Zhang ◽  
Hongyun Lu ◽  
Qihe Chen

Betulinic acid, a pentacyclic triterpene, is distributed in a variety of plants, such as birch, eucalyptus and plane trees. It shows a wide spectrum of biological and pharmacological properties, such as anti-inflammatory, antibacterial, antiviral, antidiabetic, antimalarial, anti-HIV and antitumor effects. Among them, the antitumor activity of betulinic acid has been extensively studied. However, obtaining betulinic acid from natural resources can no longer meet the needs of medicine and nutrition, so methods such as chemical synthesis and microbial biotransformation have also been used to prepare betulinic acid. At the same time, with the development of synthetic biology and genetic engineering, and the elucidation of the biosynthetic pathways of terpenoid, the biosynthesis of betulinic acid has also been extensively researched. This article reviews the preparation of betulinic acid and its pharmacological activities, in order to provide a reference for the research and utilization of betulinic acid.


Author(s):  
Albert NYARİARO ◽  
Isaac Owino OKOTH
Keyword(s):  

10.37236/8733 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Paul E. Gunnells

The Catalan numbers $C_{n} \in \{1,1,2,5,14,42,\dots \}$ form one of the most venerable sequences in combinatorics. They have many combinatorial interpretations, from counting bracketings of products in non-associative algebra to counting rooted plane trees and noncrossing set partitions. They also arise in the GUE matrix model as the leading coefficient of certain polynomials, a connection closely related to the plane trees and noncrossing set partitions interpretations. In this paper we define a generalization of the Catalan numbers. In fact we actually define an infinite collection of generalizations $C_{n}^{(m)}$, $m\geq 1$, with $C_{n}^{(1)}$ equal to the usual Catalans $C_{n}$; the sequence $C_{n}^{(m)}$ comes from studying certain matrix models attached to hypergraphs. We also give some combinatorial interpretations of these numbers.


2021 ◽  
pp. 53-55
Author(s):  
Maria Lodovica Gullino
Keyword(s):  

2021 ◽  
Vol 18 (3) ◽  
pp. 100-109
Author(s):  
Armen Ye. Petrosyan ◽  

In Old Armenian, saws means ‘proud, luxurious, great,’ ‘some (bright) color,’ and saws and sawsi mean ‘oriental plane tree’. The word has no etymology. Hurrian has the word šauša [sausa] ‘big, great’ and the theonym Šauša / Šauška for the local version of the great goddess Ištar. The article undertakes to find a single etymon looking for the clue in comparative mythology. It is known that Anušavan, one of the ancient Armenian mythical patriarchs, was referred to as Sawsanuēr which can be interpreted as “The gift of plane trees” (with a reference to the cult of the plane trees of Armawir, the earliest capital of Armenia). According to mythology, Anushavan’s father and grandfather were related to Šamiram (Greek Semiramis), the queen of the city of Nineveh (capital of Assyria) that is seen as a historicized version of the local goddess Šauš(k)a otherwise called “Ishtar of Nineveh.” The Armenian saws ‘great, magnificent’ quite correlates with this name as a loan from the Hurrian šauša ‘great,’ with a regular apocope. The plane trees were probably symbols of the goddess. Thus, it is natural to assume that the dendronym saws / sawsi (the second form with the Indo-European suffix *-iyā, characteristic of Armenian dendronyms, cf. the genitive plural form sawseac‘) is of Hurrian origin. The first meaning of the Hurrian word ‘great, magnificent’ subsequently turned into theonym and then to the Armenian dendronym, the name of the largest and most luxurious tree in the Armenian Highland and adjacent territories.


2020 ◽  
Vol 3 (12) ◽  
pp. 12627-12634
Author(s):  
Shuo Zhang ◽  
Pengcheng Dai ◽  
Haijun Liu ◽  
Liting Yan ◽  
Hexue Song ◽  
...  

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