A297565 Number of maximum matchings in the n-triangular graph.

1, 3, 8, 144, 47520, 16656192, 3321907200, 21173194506240, 7866775374741504000, 1714731229742768455680000, 149617202324844553489612800000, 1023015704130692419403265343488000000, 822671651496871196689402715812984258560000000, 267398413297417500827783894166564037306456473600000000

Offset: 2

Author

Eric W. Weisstein, Dec 31 2017

Keywords

• nonn

Links

• Eric W. Weisstein, Table of n, a(n) for n = 2..18
• Eric Weisstein's World of Mathematics, Matching
• Eric Weisstein's World of Mathematics, Maximum Independent Edge Set
• Eric Weisstein's World of Mathematics, Triangular Graph

Crossrefs

Cf. A287231, A297484.

Programs

PARI

\\ groups all orientations of n-complete graph by out degree configuration.
CompleteOrientationsByOutDegrees(n)={ \\ high memory usage and slow for n > 10
local(M=Map());
my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
my(recurse(p, i, q, v, e)=if(i<0, acc(x^e+q, v), my(t=polcoeff(p, i)); for(k=0, t, self()(p, i-1, (t-k+x*k)*x^i+q, binomial(t, k)*v, e+t-k))));
my(iterate(v, k, f)=for(i=1, k, v=f(v)); v);
iterate(Mat([1, 1]), n-1, src->M=Map(); for(i=1, matsize(src)[1], my(p=src[i, 1]); recurse(p, poldegree(p), 0, src[i, 2], 0)); Mat(M))
}
a(n)={
my(v=vector(n\2, n, (2*n)!/(2^n*n!)));
my(c(p)=my(h=(poldegree(p)+1)\2); my(r=n-1-sum(i=1, h, polcoeff(p, 2*i-1))); if(r%2, n*r/2, 1)*if(r<2, 1, v[r\2])*prod(i=1, h, v[i]^(polcoeff(p, 2*i)+polcoeff(p, 2*i-1))));
my(M=CompleteOrientationsByOutDegrees(n-1));
sum(i=1, matsize(M)[1], M[i, 2]*c(M[i, 1]))
} \\ Andrew Howroyd, Jan 02 2018

Extensions

a(10)-a(15) and offset corrected by Andrew Howroyd, Jan 02 2018

a(16)-a(18) from Eric W. Weisstein, Jan 06-08 2018