Assignments will count as 10% of the course grade, and are due by Noon on Tuesdays in the homework box for our class, by 410 Math.
Richard Stanley has written a short paper describing the correspondence between polygon dissections and Young tableaux. Find it, read it, and figure out the correspondence for the 21 ways to cut a hexagon into three pieces.
Here is some drawing paper for studying this correspondence for hexagons cut into three pieces. Fill in the Young tableau that corresponds to each dissection:
The following worksheet and example may be helpful:
Update: Here is the definitive paper establishing a direct correspondence proving the hook length formula, with references to earlier work:
Here is how Stanley’s paper appears in the bibliography of a later paper that cites him:
The easiest way to find this paper is using MathSciNet, a search engine for math papers. From a Columbia University web connection, one can simply navigate to the site. Elsewhere, there is a link via the library that requires a Columbia uni:
Search for Author “Stanley” and Title “Dissections”.
Click on the “Article” link of the search result.
On the next page, click on “Download PDF” to view and download Stanley’s paper.
Depending on your browser, it may be a puzzle to figure out how to save the file to your computer. On my browser, there’s a pop-up row of icons near the lower right corner of the page, and the second icon from the right saves the file. The icon looks like a floppy disk, an ancient form of physical computer media:
While you’re on MathSciNet, what else you can learn about polygon dissections?
It would be nice if someone set up an account at the Online Encyclopedia of Integer Sequences, and improved these entries to point to Stanley’s paper:
How many ways can one cut an n-gon into three pieces?
Number of diagonal dissections of an n-gon into 3 regions
How many ways can one cut an n-gon into four pieces?
Number of diagonal dissections of an n-gon into 4 regions