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Classical q-Numbers:  A Study of the Case q = -1

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<p>Several of the classical sequences in  enumerative combinatorics have q-generalizations  arising as generating functions for statistics  defined on finite discrete structures.  When q = 1,  these generating functions reduce to the original  sequences.  When q = -1, on the other hand, one  gets the difference in cardinalities between those  members of a set having an even value for some  statistic (on the set) with those members having an  odd value.  The current text provides a systematic  study of the  case  q = -1, giving both algebraic  and combinatorial treatments.  For the latter,  appropriate sign-reversing involutions are defined  on the associated class of discrete structures.   Among the structures studied are permutations,  binary sequences, Laguerre configurations,  derangements, Catalan words, and finite set  partitions.  As a consequence of our results, we  obtain bijective proofs of congruences involving  Stirling, Bell, and Catalan numbers. This text  studies an interesting problem in enumerative  combinatorics and is suitable for an audience  ranging from motivated undergraduates to researchers  in the field.</p><br/><p>Mark Shattuck, Ph.D., has research interests in enumerative  combinatorics and discrete probability.  He is currently a Post- doctoral Researcher at the University of Tennessee, where he  earned his doctoral degree in mathematics, specializing in  combinatorics.</p>

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