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The combinatorial relationship between trees, cacti and certain connection coefficients for the symmetric group. (English) Zbl 0804.05023
An $$m$$-cactus is a connected graph in which every edge lies on exactly one cycle, which has length $$m$$ (case $$m=2$$ corresponds to trees). If $$\sigma \in S_ n$$ is a permutation, let $$\alpha (\sigma)$$ be a partition of $$n$$, corresponding its cyclic structure, $$l(\sigma)$$ its length (number of cycles).
A combinatorial bijection between $$m$$-cacti and $$m$$-tuples $$(\sigma_ 1, \sigma_ 2, \dots, \sigma_ m)$$ such that $$\sigma_ 1 \sigma_ 2 \dots \sigma_ m = (1,2, \dots,n)$$ and $$\sum I (\sigma_ i) = n + 1$$ is established. If $$K_ \alpha = \sum_{\alpha (\sigma) = \alpha} \sigma$$ is an element of a group algebra, this bijection permits to find the exact value of the coefficient $$c^{(n)}$$ in the decomposition $$K_{\alpha_ 1} K_{\alpha_ 2} \cdots K_{\alpha_ m} = \sum_ \gamma c^ \gamma K_ \gamma$$.

##### MSC:
 05C05 Trees 05A17 Combinatorial aspects of partitions of integers 20B30 Symmetric groups
##### Keywords:
symmetric group; cactus; cycle; tree; partition; group algebra; coefficient
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