Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 50, No. 1, pp. 195213 (2009) 

The partition problem for equifacetal simplicesAllan L. EdmondsDepartment of Mathematics, Indiana University, Bloomington, IN 47405 USA, email: edmonds@indiana.eduAbstract: Associated with any equifacetal $d$simplex, which necessarily has a vertex transitive isometry group, there is a welldefined partition of $d$ that counts the number of edges of each possible length incident at a given vertex. The partition problem asks for a characterization of those partitions that arise from equifacetal simplices. The partition problem is resolved by proving that a partition of the number $d$ arises this way if and only if the number of odd entries in the partition is at most $\iota(d+1)$, the maximum number of involutions in a finite group of order $d+1$. When $n$ is even the number $\iota(n)$ is shown to be $n/2+n_{2}/21$, where $n_{2}$ denotes the $2$part of $n$. Those extremal equifacetal $d$simplices for which the number of odd entries of the associated partition is exactly $\iota(d+1)$ are characterized. Keywords: equifacetal simplex, isometry group, partition, involution Classification (MSC2000): 52B12; 52B11, 52B15, 20D60, 51N20 Full text of the article:
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