Journal of Integer Sequences, Vol. 10 (2007), Article 07.6.2

On Hultman Numbers

Jean-Paul Doignon and Anthony Labarre
Université Libre de Bruxelles
Département de Mathématique, c.p. 216
Bd du Triomphe
B-1050 Bruxelles


Finding a sequence of transpositions that transforms a given permutation into the identity permutation and is of the shortest possible length is an important problem in bioinformatics. Here, a transposition consists in exchanging two contiguous intervals of the permutation. Bafna and Pevzner introduced the cycle graph as a tool for working on this problem. In particular, they took advantage of the decomposition of the cycle graph into so-called alternating cycles. Later, Hultman raised the question of determining the number of permutations with a cycle graph containing a given quantity of alternating cycles. The resulting number is therefore similar to the Stirling number of the first kind. We provide an explicit formula for computing what we call the Hultman numbers, and give a few numerical values. We also derive formulae for related cases, as well as for a much more general problem. Finally, we indicate a counting result related to another operation on permutations called the "block-interchange".

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(Concerned with sequences A002619 A035319 and A060593 .)

Received December 23 2005; revised versions received August 21 2006; June 9 2007. Published in Journal of Integer Sequences, June 10 2007.

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