Birkhoff

The Ehrhart polynomial of the Birkhoff polytope
Matthias Beck and Dennis Pixton

Our paper: Matthias Beck and Dennis Pixton, The Ehrhart polynomial of the Birkhoff polytope, Discrete Comput. Geom., 30 (2003), 623-637. (PDF)
Our announcement of the calculation of the volume of the tenth polytope
The full Ehrhart polynomials for n ≤ 9.
The volumes of the Birkhoff polytope for n ≤ 10. The volume for n = 9 has been corrected (Dean Hickerson noticed that we forgot the scaling factor 9⁸).
Calculation times.
The source code for our program.
The On-Line Encyclopedia of Integer Sequences is a wonderful resource. It currently catalogues more than 250,000 integer sequences, many with citations and commentary, and provides a powerful search facility. For sequences related to the Birkhoff polytope see sequence A037302.

n = 10

The calculation of the volume of the Birkhoff polytope when n = 10 requires the evaluation of 4862 integrals. We started the calculation on March 13, 2002, using between 10 and 40 computers (depending on load).

The last integral was finished on May 18, 2003 The total computer time, scaled to a "standard" processor running at 1 GHz, is

6160 days, or almost 17 years.
And here's the answer:

727291284016786420977508457990121862548823260052557333386607889 / 828160860106766855125676318796872729344622463533089422677980721388055739956270293750883504892820848640000000

The Ehrhart polynomial of the Birkhoff polytope Matthias Beck and Dennis Pixton

n = 10

727291284016786420977508457990121862548823260052557333386607889 / 828160860106766855125676318796872729344622463533089422677980721388055739956270293750883504892820848640000000

The Ehrhart polynomial of the Birkhoff polytope
Matthias Beck and Dennis Pixton