A037302 Normalized volume of Birkhoff polytope of n X n doubly-stochastic square matrices. If the volume is v(n), then a(n) = ((n-1)^2)! * v(n) / n^(n-1).
1, 1, 3, 352, 4718075, 14666561365176, 17832560768358341943028, 12816077964079346687829905128694016, 7658969897501574748537755050756794492337074203099, 5091038988117504946842559205930853037841762820367901333706255223000
Offset: 1
Examples
a(2)=1: The polytope of 2 X 2 matrices is the line segment from (1,0;0,1) to (0,1;1,0), with length v(2)=2, so a(2) = 1! * 2 / 2^1 = 1.
Links
- Matthias Beck and Dennis Pixton, The Ehrhart polynomial of the Birkhoff polytope
- Matthias Beck, Stanley's Major Contributions to Ehrhart Theory, arXiv preprint arXiv:1407.0255 [math.CO], 2014.
- Matthias Beck and Dennis Pixton, The Ehrhart polynomial of the Birkhoff polytope, arXiv:math/0202267 [math.CO], 2002-2005.
- Matthias Beck and Dennis Pixton, The Ehrhart polynomial of the Birkhoff polytope, Discrete Comput. Geom. 30 (2003), no. 4, 623-637.
- Petter Brändén, Jonathan Leake, and Igor Pak, Lower bounds for contingency tables via Lorentzian polynomials, arXiv:2008.05907 [math.CO], 2020.
- C. S. Chan and D. P. Robbins, On the volume of the polytope of doubly stochastic matrices, arXiv:math/9806076 [math.CO], 1998.
- C. S. Chan and D. P. Robbins, On the volume of the polytope of doubly stochastic matrices, Exper. Math. 8 (1999), 291-300.
- Jesús A. De Loera, Fu Liu, and Ruriko Yoshida, A generating function for all semi-magic squares and the volume of the Birkhoff polytope, J. Algebraic Combin. 30 (2009), no. 1, 113-139.
- R. P. Stanley, Decompositions of rational convex polytopes, Annals of Discrete Math. 6 (1980), 333-342.
Crossrefs
Formula
Extensions
v(9) computed by Matthias Beck (matthias(AT)math.binghamton.edu) and Dennis Pixton (dennis(AT)math.binghamton.edu), Feb 25 2002
Edited by Dean Hickerson, Nov 27 2002
a(10) is based on a calculation of v(10) by Matthias Beck (matthias(AT)math.binghamton.edu) and Dennis Pixton (dennis(AT)math.binghamton.edu) from Mar 13 2002 to May 18 2003
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