A078525 -id:A078525 - 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).

1, 1, 3, 352, 4718075, 14666561365176, 17832560768358341943028, 12816077964079346687829905128694016, 7658969897501574748537755050756794492337074203099, 5091038988117504946842559205930853037841762820367901333706255223000

Offset: 1

Günter M. Ziegler (ziegler(AT)math.tu-berlin.de)

The Birkhoff polytope is an (n-1)^2-dimensional polytope in n^2-dimensional space; its vertices are the n! permutation matrices.

Is a(n) divisible by n^2 for all n>=4? - Dean Hickerson, Nov 27 2002

			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.

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.

Numerator and denominator of v(n) are in A078524 and A078525.
Row sums of A259473.
Cf. A257493.

a(n) = ((n-1)^2)!*A078524(n)/(n^(n-1)*A078525(n)). - Andrew Howroyd, Apr 11 2020

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

A078524 Numerator of volume of Birkhoff polytope of n X n doubly-stochastic square matrices.

Original entry on oeis.org

1, 2, 9, 176, 23590375, 9700106723, 77436678274508929033, 5562533838576105333259507434329, 559498129702796022246895686372766052475496691, 727291284016786420977508457990121862548823260052557333386607889

Offset: 1

Dean Hickerson, Nov 27 2002

			Volumes for n=1 to 10, from Beck-Pixton (2003):
1 1
2 2
3 9/8
4 176/2835
5 23590375/167382319104
6 9700106723/1319281996032000000
7 77436678274508929033/13730296368223523839986892800000000
8 5562533838576105333259507434329/1258903626009547795008148094269333980330892800
00000000
9 559498129702796022246895686372766052475496691/9269262340995263649896514671280698429605195132920241960610847715334553600000000000000
10 727291284016786420977508457990121862548823260052557333386607889/828160860106766855125676318796872729344622463533089422677980721388055739956270293750883504892820848640000000

N. J. A. Sloane, Table of n, a(n) for n = 1..10
Beck, Matthias, Stanley's Major Contributions to Ehrhart Theory, arXiv preprint arXiv:1407.0255 (2014).
Matthias Beck and Dennis Pixton, The Ehrhart polynomial of the Birkhoff polytope, Discrete Comput. Geom. 30 (2003), no. 4, 623-637, arXiv:math.CO/0202267

Denominator is in A078525. See A037302 for more information about volume of Birkhoff polytopes.

Entry revised by N. J. A. Sloane, Apr 16 2016

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).

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