Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #26 Nov 01 2019 12:34:49
%S 1,2,7,34,221,1736,15584,153228,1611189,17826202,205282376,2441437708,
%T 29816628471,372314544202,4737438631001,61264426341926,
%U 803488037899349,10668478221202710,143203795004873285,1940953294927992976,26536578116407809962,365653739580163294032
%N Number of directed (or Gale-Ryser) graphical partitions: degree-vector pairs (in-degree, out-degree) for directed graphs (loops allowed) with n vertices; or possible ordered pair (row-sum, column-sum) vectors for a 0-1 matrix.
%D R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.
%H Andrew Howroyd, <a href="/A029894/b029894.txt">Table of n, a(n) for n = 0..30</a>
%H Peter L. Erdos, I Miklós, Z Toroczkai, <a href="http://arxiv.org/abs/1601.08224">New classes of degree sequences with fast mixing swap Markov chain sampling</a>, arXiv preprint arXiv:1601.08224 [math.CO], 2016.
%H <a href="/index/Gra#graph_part">Index entries for sequences related to graphical partitions</a>
%F Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.
%F a(n) = F(n, n, 0, n) where F(b, c, t, w) = Sum_{i=0..b} Sum_{j=ceiling((t+i)/w))..min(t+i, c)} F(i, j, t+i-j, w-1) for w > 0, F(b, c, 0, 0) = 1 and F(b, c, t, 0) = 0 for t > 0. - _Andrew Howroyd_, Nov 01 2019
%o (PARI) \\ see A327913 for T(n,m)
%o for(n=0, 15, print1(T(n,n), ", ")) \\ _Andrew Howroyd_, Nov 01 2019
%Y Main diagonal of A327913.
%Y Cf. A000569, A004250, A004251, A029889, A318396.
%K nonn
%O 0,2
%A torsten.sillke(AT)lhsystems.com
%E "Loops allowed" added to the definition by _Brendan McKay_, Oct 20 2015
%E a(0)=1 prepended and terms a(12) and beyond from _Andrew Howroyd_, Oct 31 2019