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
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.
A078525
Denominator of volume of Birkhoff polytope of n X n doubly-stochastic square matrices.
Original entry on oeis.org
1, 1, 8, 2835, 167382319104, 1319281996032000000, 13730296368223523839986892800000000, 125890362600954779500814809426933398033089280000000000
Offset: 1
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.
A337650
Volume of the positive signed Birkhoff polytope BB_{+}(n).
Original entry on oeis.org
1, 4, 642, 12065248, 53480547965190
Offset: 1
- Dylan Heuer and Jessica Striker, Partial Permutation and Alternating Sign Matrix Polytopes, SIAM Journal on Discrete Mathematics, 36 (2022), 2863-2888; arXiv:2012.09901 [math.CO], 2020-2022. See the diagonal of the table in Fig. 1.
- Florian Kohl, McCabe Olsen, and Raman Sanyal, Unconditional reflexive polytopes, arXiv:1906.01469 [math.CO], 2019. Also Discrete and Computational Geom., 64:2 (2020), 427-452.
A259473
Irregular triangle read by rows of coefficients arising in the enumeration of doubly stochastic matrices of integers, n >= 1, 0 <= k <= (n-1)*(n-2).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 14, 87, 148, 87, 14, 1, 1, 103, 4306, 63110, 388615, 1115068, 1575669, 1115068, 388615, 63110, 4306, 103, 1, 1, 694, 184015, 15902580, 567296265, 9816969306, 91422589980, 490333468494, 1583419977390, 3166404385990, 3982599815746, 3166404385990
Offset: 1
Triangle begins:
1;
1;
1,1,1;
1,14,87,148,87,14,1;
1,103,4306,63110,388615,1115068,1575669,1115068,388615,63110,4306,103,1;
...
- Andrew Howroyd, Table of n, a(n) for n = 1..177 (rows 1..9)
- D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4 (1975), 474-477, doi:10.1137/0204040.
- D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
a(1)=1 prepended and terms a(26) and beyond from
Andrew Howroyd, Apr 11 2020
Showing 1-4 of 4 results.
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