A351869 a(n) is the number of self-complementary score sequences that are possible for strong tournaments on n vertices.

1, 1, 0, 1, 1, 3, 3, 9, 11, 30, 39, 103, 141, 363, 514, 1301, 1894, 4727, 7036, 17358, 26311, 64282, 98936, 239712, 373806, 899115, 1418130, 3389078, 5399133, 12828749, 20619565, 48739465, 78963217, 185769110, 303128971, 710067027, 1166206802, 2720959217, 4495461790

Offset: 0

Author

Paul K. Stockmeyer, Feb 22 2022

Keywords

• nonn

Comments

See A000571 for the definition of a tournament and its score sequence. A tournament is strong if every two vertices are mutually reachable by directed paths. Alternatively, a tournament is strong if it contains a directed Hamiltonian cycle.

A sequence (s_1 <= s_2 <= ... <= s_n) of integers is the score sequence of a strong tournament iff Sum_{i=1..r} s_i > binomial(r,2) for 1 <= r < n and Sum_{i=1..n} s_i = binomial(n,2). It is self-complementary iff s_{n+1-i} = n-1-s_i for 1 <= i <= n/2.

Examples

			The 9 self-complementary strong score sequences of length seven are
  (1,1,2,3,4,5,5),
  (1,1,3,3,3,5,5),
  (1,2,2,3,4,4,5),
  (1,2,3,3,3,4,5),
  (1,3,3,3,3,3,5),
  (2,2,2,3,4,4,4),
  (2,2,3,3,3,4,4),
  (2,3,3,3,3,3,4),
  (3,3,3,3,3,3,3).
		

Links

• Paul K. Stockmeyer, Table of n, a(n) for n = 0..501
• Paul K. Stockmeyer, Counting Various Classes of Tournament Score Sequences, arXiv:2202.05238 [math.CO], 2022.
• Paul K. Stockmeyer, Counting Various Classes of Tournament Score Sequences, J. Integer Seq. 26 (2023), Article 23.5.2.

Crossrefs

Cf. A000571, A351822, A345470.

Formula

For n >= 1, a(2*n) = Sum_{T=binomial(n,2)+1..n*(n-1)} Sum_{E=floor(n/2)..n-1} g_n(T,E) and a(2*n+1) = Sum_{T=binomial(n,2)+1..n^2} Sum_{E=floor(n/2)..n} g_n(T,E), where g_1(T, E)=[T=E], and for n>=2, g_n(T, E)=0 if T-E <= binomial(n-1, 2) and g_n(T, E) = Sum_{k=0..E} g_{n-1}(T-E, k) otherwise.