A345470 Number of self-complementary score sequences that are possible in an n-team round-robin tournament.

1, 1, 1, 2, 2, 5, 6, 15, 19, 48, 64, 161, 223, 557, 796, 1971, 2887, 7090, 10596, 25826, 39256, 95016, 146533, 352411, 550328, 1315827, 2077418, 4940587, 7876036, 18639221, 29971423, 70608885, 114422037, 268436473, 438068242

Offset: 0

Author

Howard Givner, Jun 20 2021

Keywords

• nonn

Comments

See A000571 for the definition of a score sequence.

A self-complementary score sequence W is a score sequence of win counts such that W = {s(1), s(2), ..., s(n)} and its complement, L={n-1-s(n), n-1-s(n-1), ..., n-1-s(1)}, a score sequence of loss counts, are identical.

Examples

			For n = 4 there are 4 score sequences of which only 2, those marked with an asterisk, are self-complementary.  These are the sequences for n=4.
    {0,1,2,3} *
    {0,2,2,2}
    {1,1,1,3}
    {1,1,2,2} *
For n = 5, there are 9 score sequences of which only 5, those marked with an asterisk, are self-complementary.  These are the sequences for n=5.
    {0,1,2,3,4} *
    {0,1,3,3,3}
    {0,2,2,2,4} *
    {0,2,2,3,3}
    {1,1,2,3,3} *
    {1,1,1,3,4}
    {1,1,2,2,4}
    {1,2,2,2,3} *
    {2,2,2,2,2} *
		

Links

• Paul K. Stockmeyer, Table of n, a(n) for n = 0..500
• Paul K. Stockmeyer, Counting Various Classes of Tournament Score Sequences, J. Integer Seq. 26 (2023), Article 23.5.2.

Crossrefs

Cf. A000571.

Extensions

a(30) corrected by Howard Givner, Jun 28 2021

a(0)=1 prepended and a(1) changed from 0 to 1 by Howard Givner, Feb 22 2022