I've read about some studies on the Paley I Construction. Among them I found the following notations ( See this page: https://documents.uow.edu.au/~jennie/matrices/32P02.html ).


I am not familiar with these notations. Could you please recommend me some books or articles about it? Thanks in advance!

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    $\begingroup$ Smith Normal Form, my first guess, as in the answer. $\endgroup$
    – Will Jagy
    Apr 21 at 14:15
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    $\begingroup$ If you want the very basics about some topic in math, Wikipedia is always a good place to check: en.wikipedia.org/wiki/Smith_normal_form $\endgroup$ Apr 21 at 15:36
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    $\begingroup$ To complete the guess of @Will Jagy, in your example the SNF is given by the diagonal matrix $\operatorname{diag}(1,2^a,4^b,8^b,16^a,32,0,\ldots)$ for adequate non-negative integers $a,b$, where for each diagonal entry $d_i$ divides $d_{i+1}$. You can find additional details on the same page where you found the example: documents.uow.edu.au/~jennie/WEBPDF/1996_08.pdf $\endgroup$
    – F Zaldivar
    Apr 21 at 17:18
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    $\begingroup$ Brouwer and Haemers' "Spectra of Graphs" has a chapter about p-ranks, it is Chapter 13. In the last section of that chapter they discuss SNFs. Not the most basic reference but I think more in line with the type of SNF you are interested in, as a graph invariant. win.tue.nl/~aeb/2WF05/spectra.pdf By the way, the exponents a, b in your example are meant to indicate the multiplicity of the entry. So "2" occurs a times. $\endgroup$
    – Josh
    Apr 22 at 19:24

1 Answer 1


I think Richard Stanley's survey would be a good start. The published version is here.

Richard also has some slides regarding SNF.


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