Higher order SPT functions for overpartitions, overpartitions with smallest part even, and partitions with smallest part even and without repeated odd parts

We consider the symmetrized moments of three ranks and cranks, similar to the work of Garvan in [17] for the rank and crank of a partition. By using Bailey pairs and elementary rearrangements, we are able to find useful expressions for these moments. We then deduce inequalities between the corresponding ordinary moments. In particular we prove that the crank moment for overpartitions is always larger than the rank moment for overpartitions, M¯2k(n)>N¯2k(n); with recent asymptotics this was known to hold for sufficiently large values of n for each fixed k. Lastly we provide higher order spt functions for overpartitions, overpartitions with smallest part even, and partitions with smallest part even and no repeated odds.