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In this arXiv paper (p. 13), Steinhardt defines a normal set in $S_n$ as follows:

Definition: A split set of more than two cycles generating $S_n$ is said to be normal if any element is adjacent to at most 1 leaf, and furthermore $Cyc(T)$ is a tree.

My question is on the above definition. I'm not sure what the "adjacent" refers to. Is this an adjacency relation in some graph, and if so, which graph? Could you illustrate this definition with some examples of normal sets $T$ in $S_n$ for $n=5,6,7$?

Let me recall from the paper the following definitions. A set of cycles $T$ is said to be split if the intersection of the supports of any two elements in $T$ has size at most 1. Given a split set of cycles $T \subset S_n$, $Cyc(T)$ is defined to be the graph with vertex set $\{1,\ldots,n\}$ and with edges $(x_1,x_2),\ldots,(x_{k-1},x_k)$ whenever $(x_1,\ldots,x_k)$ is a cycle in $T$ (here $x_1$ needs to be chosen arbitrarily). The degree of $t \in T$ is the number of distinct points in its support that overlap with other cycles in $T$. An element $t \in T$ is a leaf it its degree is 1.

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  • $\begingroup$ It seems to me that $\operatorname{Cyc}(T)$ depends on the choices of the starting points $x_1$ for each cycle, and thus is not well-defined. At least, one has to show that whether $\operatorname{Cyc}(T)$ is a tree or not, is independent of these choices. I could imagine that two elements of $T$ are called "adjecent" if their supports have nonempty intersection, but this is just a guess. $\endgroup$
    – Frieder Ladisch
    Commented Mar 10, 2016 at 13:56
  • $\begingroup$ @FriederLadisch Yes, I guessed the same. $\endgroup$
    – svsring
    Commented Mar 10, 2016 at 15:56

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