I submitted my application for the National Science Foundation graduate fellowship at around 7:00 this morning. This nearly ends an extremely hectic week, likely to be the most hectic of the quarter. The only thing remaining is the dreadful GRE subject test tomorrow. I am rewarding myself by going to hear Musica Antiqua Köln perform Heinichen, Zelenka, and Telemann on Sunday. It will be interesting to see which way Reinhard Goebel plays the violin nowadays.

I have been enjoying my combinatorics class a lot lately. If nothing else, it has been providing evidence that I have actually learned something in college. (That’s a good thing for a senior, right?) But posets are also fun on their own. We were talking about incidence algebras yesterday, and they’re really interesting. My intuition says that incidence algebras essentially tell you everything about a poset that a Dirichlet series can tell you about the integers, especially if we represent the integers by an appropriate Hasse diagram. (That’s probably just about everything, for the record.) For example, if you want to know the number of length k chains between two elements of a poset, I can easily write down an element of the incidence algebra that will tell you that immediately.

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## About Simon

Hi. I'm Simon Rubinstein-Salzedo. I'm a mathematics postdoc at Dartmouth College. I'm also a musician; I play piano and cello, and I also sometimes compose music and study musicology. I also like to play chess and write calligraphy. This blog is a catalogue of some of my thoughts. I write them down so that I understand them better. But sometimes other people find them interesting as well, so I happily share them with my small corner of the world.

Is there any relationship between the incidence algebra of a graded poset and the incidence matrices you get between any two consecutive ranks from the Hasse diagram? Convolutions and matrix multiplication appear to have a lot in common, so I’m curious if these two methods of analysis produce any of the same kinds of results.

If I understand correctly what you mean by the incidence matrices you get between any two consecutive ranks of the Hasse diagram, then the incidence algebra should be generated by (I-incidence matrices with diagonal entries not equal to 1)^{-1} for a finite graded poset. The incidence algebra is the algebra of upper triangular matrices that have zero ij-entry unless i<=j. So you can write this as a sum of identity relations, covering relations, squares of covering relations, and so forth uniquely when you have a graded poset. (You can do that when it's not graded as well, but then it won't be unique.)