A029894 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.

1, 2, 7, 34, 221, 1736, 15584, 153228, 1611189, 17826202, 205282376, 2441437708, 29816628471, 372314544202, 4737438631001, 61264426341926, 803488037899349, 10668478221202710, 143203795004873285, 1940953294927992976, 26536578116407809962, 365653739580163294032

Offset: 0

Author

torsten.sillke(AT)lhsystems.com

Keywords

• nonn

References

• R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..30
• Peter L. Erdos, I Miklós, Z Toroczkai, New classes of degree sequences with fast mixing swap Markov chain sampling, arXiv preprint arXiv:1601.08224 [math.CO], 2016.
• Index entries for sequences related to graphical partitions

Crossrefs

Main diagonal of A327913.

Cf. A000569, A004250, A004251, A029889, A318396.

Programs

PARI

\\ see A327913 for T(n,m)
for(n=0, 15, print1(T(n,n), ", ")) \\ Andrew Howroyd, Nov 01 2019

Formula

Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.

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

Extensions

"Loops allowed" added to the definition by Brendan McKay, Oct 20 2015

a(0)=1 prepended and terms a(12) and beyond from Andrew Howroyd, Oct 31 2019