TY - JOUR

T1 - Combinatorial game distributions of steiner systems

AU - Irie, Yuki

N1 - Funding Information:
∗This work was partially supported by JSPS KAKENHI Grant Number JP20K14277. The paper is a substantially revised version of Section 2.A of the author’s Ph.D. thesis, which was prepared under the supervision of Masaaki Kitazume for Chiba University. This revision includes adding a result that characterizes projective Steiner triple systems.
Publisher Copyright:
© The author.

PY - 2021

Y1 - 2021

N2 - The P-position sets of some combinatorial games have special combinatorial structures. For example, the P-position set of the hexad game, first investigated by Conway and Ryba, is the block set of the Steiner system S(5, 6, 12) in the shuffle numbering, denoted by Dsh . However, few games were known to be related to Steiner systems in this way. For a given Steiner system, we construct a game whose P-position set is its block set. By using constructed games, we obtain the following two results. First, we characterize Dsh among the 5040 isomorphic S(5, 6, 12) with point set {0, 1, …, 11}. For each S(5, 6, 12), our construction produces a game whose P-position set is its block set. From Dsh, we obtain the hexad game, and this game is characterized as the unique game with the minimum number of positions among the obtained 5040 games. Second, we characterize projective Steiner triple systems by using game distributions. Here, the game distribution of a Steiner system D is the frequency distribution of the numbers of positions in games obtained from Steiner systems isomorphic to D. We find that the game distribution of an S(t, t + 1, v) can be decomposed into symmetric components and that a Steiner triple system is projective if and only if its game distribution has a unique symmetric component.

AB - The P-position sets of some combinatorial games have special combinatorial structures. For example, the P-position set of the hexad game, first investigated by Conway and Ryba, is the block set of the Steiner system S(5, 6, 12) in the shuffle numbering, denoted by Dsh . However, few games were known to be related to Steiner systems in this way. For a given Steiner system, we construct a game whose P-position set is its block set. By using constructed games, we obtain the following two results. First, we characterize Dsh among the 5040 isomorphic S(5, 6, 12) with point set {0, 1, …, 11}. For each S(5, 6, 12), our construction produces a game whose P-position set is its block set. From Dsh, we obtain the hexad game, and this game is characterized as the unique game with the minimum number of positions among the obtained 5040 games. Second, we characterize projective Steiner triple systems by using game distributions. Here, the game distribution of a Steiner system D is the frequency distribution of the numbers of positions in games obtained from Steiner systems isomorphic to D. We find that the game distribution of an S(t, t + 1, v) can be decomposed into symmetric components and that a Steiner triple system is projective if and only if its game distribution has a unique symmetric component.

UR - http://www.scopus.com/inward/record.url?scp=85121218048&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85121218048&partnerID=8YFLogxK

U2 - 10.37236/9252

DO - 10.37236/9252

M3 - Article

AN - SCOPUS:85121218048

VL - 28

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 4

M1 - P4.54

ER -