Block number, descents and Schur positivity of fully commutative elements in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e293" altimg="si126.svg"><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>

The distribution of Coxeter descents and block number over the set fully commutative elements in hyperoctahedral group B n , FC ( ) is studied this paper. We prove that associated Chow quasi-symmetric generating function equal to a non-negative sum products two Schur functions. proof involves decomposition into disjoint union two-sided Barbash–Vogan combinatorial cells, type extension Rubey’s descent preserving involution on 321-avoiding permutations detailed study intersection with S -cosets which yields new subsets called fibers. also compare different Schur-positivity notions, arising from works Poirier.