OFFSET
1,3
COMMENTS
For any D with n vertices, Var(X_m(D)) is either identically zero or a polynomial in m of degree between n and 2*n-3.
It follows from Theorem 6 in Andersson (1998) that a(n) = 0 if and only if n is a power of 2.
The expected value of X_m(D) equals m!/((m-n)!*2^e*aut(D)), where e is the number of edges of D and aut(D) is the number of automorphisms of D.
If D and D' are the reverses of each other, i.e., D' is obtained from D by reversing the direction of all its edges, X_m(D) and X_m(D') are equidistributed.
LINKS
Pontus Andersson (von Brömssen), On the asymptotic distributions of subgraph counts in a random tournament, Random Structures & Algorithms 13 (1998), 249-260.
Pontus Andersson (von Brömssen), Small variance of subgraph counts in a random tournament, Statistics & Probability Letters 49 (2000), 135-138.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Pontus von Brömssen, Jun 18 2024
STATUS
approved