Hasse diagrams of posets with up to 7 elements, and the number of posets with 10 elements, without the use of computer programs

2017/10/27 by Monteiro, Luiz F., Savini, Sonia, Viglizzo, Ignacio
#06A07 #Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1710.10343

Abstract

Let P(n) be the set of all posets with n elements. Let P^(j)(n), 1≤ j≤ 2ⁿ, be the number of all posets with n elements possessing exactly j antichains. We have determined the numbers P^(j)(7), 1≤ j≤ 128, and using a result of M.~Erné [Erné, M., On the cardinalities of finite topologies and the number of antichains in partially ordered sets, Discrete Mathematics 35 (1981), 119-133.], we compute |P(10)| without the aid of any computer program. We include the Hasse diagrams of all the non-isomorphic posets of P(7). We also present formulas for the number of connected posets of certain forms, and use them to compute |P(n)| with 1≤ n≤ 8 by a different method.

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