cp's OEIS Frontend

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).

%I A037302 #50 Nov 15 2022 12:41:31
%S A037302 1,1,3,352,4718075,14666561365176,17832560768358341943028,
%T A037302 12816077964079346687829905128694016,
%U A037302 7658969897501574748537755050756794492337074203099,5091038988117504946842559205930853037841762820367901333706255223000
%N 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).
%C A037302 The Birkhoff polytope is an (n-1)^2-dimensional polytope in n^2-dimensional space; its vertices are the n! permutation matrices.
%C A037302 Is a(n) divisible by n^2 for all n>=4? - _Dean Hickerson_, Nov 27 2002
%H A037302 Matthias Beck and Dennis Pixton, <a href="https://matthbeck.github.io/birkhoff/">The Ehrhart polynomial of the Birkhoff polytope</a>
%H A037302 Matthias Beck, <a href="http://arxiv.org/abs/1407.0255">Stanley's Major Contributions to Ehrhart Theory</a>, arXiv preprint arXiv:1407.0255 [math.CO], 2014.
%H A037302 Matthias Beck and Dennis Pixton, <a href="https://arxiv.org/abs/math/0202267">The Ehrhart polynomial of the Birkhoff polytope</a>, arXiv:math/0202267 [math.CO], 2002-2005.
%H A037302 Matthias Beck and Dennis Pixton, <a href="https://doi.org/10.1007/s00454-003-2850-8">The Ehrhart polynomial of the Birkhoff polytope</a>, Discrete Comput. Geom. 30 (2003), no. 4, 623-637.
%H A037302 Petter Brändén, Jonathan Leake, and Igor Pak, <a href="https://arxiv.org/abs/2008.05907">Lower bounds for contingency tables via Lorentzian polynomials</a>, arXiv:2008.05907 [math.CO], 2020.
%H A037302 C. S. Chan and D. P. Robbins, <a href="https://arxiv.org/abs/math/9806076">On the volume of the polytope of doubly stochastic matrices</a>, arXiv:math/9806076 [math.CO], 1998.
%H A037302 C. S. Chan and D. P. Robbins, <a href="http://projecteuclid.org/euclid.em/1047262409">On the volume of the polytope of doubly stochastic matrices</a>, Exper. Math. 8 (1999), 291-300.
%H A037302 Jesús A. De Loera, Fu Liu, and Ruriko Yoshida, <a href="https://www.emis.de/journals/JACO/Volume30_1/m6627810x2013373.html">A generating function for all semi-magic squares and the volume of the Birkhoff polytope</a>, J. Algebraic Combin. 30 (2009), no. 1, 113-139.
%H A037302 R. P. Stanley, <a href="https://doi.org/10.1016/S0167-5060(08)70717-9">Decompositions of rational convex polytopes</a>, Annals of Discrete Math. 6 (1980), 333-342.
%F A037302 a(n) = ((n-1)^2)!*A078524(n)/(n^(n-1)*A078525(n)). - _Andrew Howroyd_, Apr 11 2020
%e A037302 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.
%Y A037302 Numerator and denominator of v(n) are in A078524 and A078525.
%Y A037302 Row sums of A259473.
%Y A037302 Cf. A257493.
%K A037302 nonn,hard,nice
%O A037302 1,3
%A A037302 Günter M. Ziegler (ziegler(AT)math.tu-berlin.de)
%E A037302 v(9) computed by Matthias Beck (matthias(AT)math.binghamton.edu) and Dennis Pixton (dennis(AT)math.binghamton.edu), Feb 25 2002
%E A037302 Edited by _Dean Hickerson_, Nov 27 2002
%E A037302 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