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Ranks and subdegrees of the symmetric group \(S_n\) acting on unordered \(r\)-element subsets. (English) Zbl 1399.05226

Summary: The main aim of this paper is to determine the ranks and subdegrees of the symmetric group \(S_n\) acting on unordered \(r\)-element subsets of \(X = \{1, 2, 3,\ldots, n\}\). These areas have not received much attention, in fact most of the research has been focused on the action of \(S_n\) on unordered pairs. In this paper it has been shown that the action of \(S_n\) on unordered pairs. In this paper it has been shown that the action of \(S_n\) on \(X^{(r)}\) is transitive. The ranks and suborbits of \(S_n\) acting on \(X^{(4)}\) and \(X^{(5)}\) are determined, after which it is proved that the rank of \(S_n\) acting on \(X^{(r)}\) is \(r+1\) if \(n \leq 2r\). It has been shown that the suborbits of \(S_n\) acting on \(X^{(r)}\) are self paired. It is also proved thet the subdegrees of \(S_n\) actiong on \(X^{(r)}\) are \(1,4,\binom{n-4}{r-1}, \binom{r}{2}\binom{n-r}{r-2},\binom{r}{3}\binom{n-r}{r-3},\dots,\binom{r}{r-1}\binom{n-r}{1}, \binom{n-r}{r}\).

MSC:

05E10 Combinatorial aspects of representation theory
20B30 Symmetric groups
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