Abstract
Several signed excedance-like statistics have nice formulae or generating functions when summed over the symmetric group and over its subset of derangements. We give counterparts of some of these results when we sum over the hyperoctahedral group and its subset of derangements. Our results motivate us to define and derive attractive bivariate formulae which generalise some of these results for the symmetric group.
Highlights
Several signed excedance-like statistics have nice formulae or generating functions when summed over the symmetric group and over its subset of derangements
We give counterparts of some of these results when we sum over the hyperoctahedral group and its subset of derangements
Our results motivate us to define and derive attractive bivariate formulae which generalise some of these results for the symmetric group
Summary
For a positive integer n 1, define the signed excedance enumerator as SgnExcn(q) = π∈Sn(−1)invA(π)qexc(π). An alternate proof of both these results was given by Sivasubramanian in [5], where both SgnExcn(q) and DSgnExcn(q) were found out to be determinants of suitably defined n × n matrices. It would be very interesting if the results in this paper on Bn can be obtained by evaluating the determinant of a 2n × 2n matrix, or even the determinant of a matrix with order polynomial in n, as done in [5] Our proof of these results motivate us to define bivariate signed excedance enumerators in Bn and in Sn as well. In Subsection 5.1, we show that a very similar binomial type equation is satisfied by our signed bivariate analogue when the exponent of t in the term corresponding to σ is either pos n(σ) or pos 1(σ), see Corollaries 25 and 26
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