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{\bf Farzin Barekat and Stephanie van Willigenburg}
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{\bf Composition of Transpositions and Equality of Ribbon Schur $Q$-Functions}
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We introduce a new operation on skew diagrams called composition of
transpositions, and use it and a Jacobi-Trudi style formula to derive
equalities on skew Schur $Q$-functions whose indexing shifted skew
diagram is an ordinary skew diagram. When this skew diagram is a
ribbon, we conjecture necessary and sufficient conditions for equality
of ribbon Schur $Q$-functions. Moreover, we determine all relations
between ribbon Schur $Q$-functions; show they supply a
${\Bbb Z}$-basis for skew Schur $Q$-functions; assert their
irreducibility; and show that the non-commutative analogue of ribbon
Schur $Q$-functions is the flag $h$-vector of Eulerian posets.



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