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We define a new statistic on the even hyperoctahedral groups which is anatural analogue of the odd length statistic recently defined and studied onCoxeter groups of types $A$ and $B$. We compute the signed (by length)generating function of this statistic over the whole group and over its maximaland some other quotients and show that it always factors nicely. We alsopresent some conjectures.

Odd length for even hyperoctahedral groups and signed generating functions