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Generator sets for the alternating group. (English) Zbl 1295.05269
Summary: Although the alternating group is an index 2 subgroup of the symmetric group, there is no generating set that gives a Coxeter structure on it. Various generating sets were suggested and studied by Bourbaki, Mitsuhashi, Regev and Roichman, Vershik and Vserminov, and others. In a recent work of F. Brenti et al. [J. Comb. Theory, Ser. A 115, No. 5, 845–877 (2008; Zbl 1211.20035)], it is explained that palindromes in Mitsuhashi’s generating set play a role similar to that of reflections in a Coxeter system.

We study in detail the length function with respect to the set of palindromes. Results include an explicit combinatorial description, a generating function, and an interesting connection to Broder’s restricted Stirling numbers.

MSC:
05E15 Combinatorial aspects of groups and algebras (MSC2010)
20B30 Symmetric groups
20F05 Generators, relations, and presentations of groups
20F55 Reflection and Coxeter groups (group-theoretic aspects)
Software:
OEIS
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