# What does 合痕类的个数 (hé hén lèi de gè shù) mean in this mathematics article?

From the mathematics article 拉丁阵的枚举和计数(Ⅱ):情形n=4,k〈=4, we have the following first paragraph: I'm confused about what 合痕类的个数 (hé hén lèi de gè shù) means here (it's used twice). Google Translate says it means "the number of traces", but I don't feel this is accurate.

Question: What does 合痕类的个数 (hé hén lèi de gè shù) mean in this mathematics article?

My transcription:

My translation:

In document  is the fascinating引人 (short for 引人入胜 (?)) concept概念 of the (n,k)-Latin matrix, and the case情形 of n=2,3, gives (n,k)-Latin matrices and their [合痕类的个数 (?)]. In this paper, we continue继续 the discussion讨论 to the case情形 of n=4, give (4,k,1)-Latin matrices, (4,2), (4,3), and (4,4)-Latin matrices and their [合痕类的个数 (?)].

If it helps, here's an example of what one looks like: • search web for 合痕类, get e.g. 2001 Journal of Beijing University of Posts and Telecommunications Vol. 24 No. 1 文章编号 : 1007-5321(2001) 01-0028-04 拉丁方合痕分类的快速算法 An Efficient Algorithm of Latin Squares Isotopy Classfication 合痕 isotopy 合痕类 isotopy class, also bkrs 合痕 isotopy,合痕分类(的算法) isotopy classification – user6065 Jun 3 '18 at 5:01

I am absolutely not a mathematician, but here is my stab:

isotropic: "having the same properties in all directions,"

In your case, a regular matrix, your example showing the vertical and horizontal elements to be the same.

This document goes on to discuss n=4 matrices

showing (4,k,1) - Latin matrices, (4,2), (4,3)- and
(4,4)-拉丁阵及它们的合痕类的个数。
4 by 4 Latin matrices and their isotropic elements

• Thanks! (It's going to be "isotopic" rather than "isotropic", but I get the point.) So it looks like 个数 is used to mean "representative" (one from each isotopy class). – Becky 李蓓 Jun 3 '18 at 23:23
• bkrs: isotropy; isotropism 各向同性 (一物质在各方向具有相同物理性质的特性),also 各项同性 examples from iciba: .在这儿,我们给出适用于各项同性固体的方程.为了简单起见,仅讨论各向同性材料.re 合痕 also see baidu: 嵌入合痕 isotopy of embeddings，Q mentioned the term "trace" as supplied by google, users note that the mathematical term "trace" agrees with the general term 迹(繁体：跡) (see Wikipedia,e.g.) – user6065 Jun 4 '18 at 1:44
• 'isotope' is a word first used in 1913, suggested by the Scottish writer and doctor Margaret Todd from the Greek 'iso' = equal, same, 'topos' = place. The idea is, different atoms of the same element may have different atomic weights, but they are still 'in the same place in the periodic table'. To my mind, this clearly belongs to the field of chemistry. If mathematicians have commandeered the word and somehow applied it to maths, I would not know. My definition of isotopy would be: the phenomenon of having isotopes. It may be easy to confound isotopy and isotropy. – Pedroski Jun 4 '18 at 3:43
• I am a research mathematician. Both "isotopy/isotopic" and "isotropy/isotropic" are established mathematical terms in English, with completely different meanings. However I am not sure which would be used for Latin squares (though I'd guess "isotropic", given that "isotropic subspace" arises in symplectic linear algebra). – user19168 Jun 5 '18 at 5:16