  We consider ribbon shapes, not necessarily connected, whose rows, with at
least two boxes in each, are in monotone length order. These ribbons are
uniquely defined by a pair of partitions: the row partition consisting of the
row lengths in decreasing order, and the overlapping partition whose entries
count the total number of columns with two boxes in the successive ribbon
shapes obtained by sequentially subtracting the longest row. The support of
such ribbon Schur functions, considered as a subposet of the dominance order
lattice on partitions, has the row partition as bottom element, and, as top
element, the partition whose two parts consist of the total number of columns,
and the total number of columns of length two respectively. We give a complete
system of linear inequalities in terms of the partition pair defining the
aforesaid ribbon shape under which the ribbon Schur function attains all the
Schur interval when expanded in the basis of Schur functions. We then conclude
that the Gaetz-Hardt-Sridhar necessary condition for a connected ribbon to have
full equivalence class is equivalent to the condition for a monotone connected
ribbon to have full Schur support. That is, the set of partitions with full
equivalence class is a subset of those monotone connected ribbons with full
Schur support. M. Gaetz, W. Hardt and S. Sridhar conjectured that the necessary
condition is also sufficient which translates now to every monotone connected
ribbon with full Schur support has full equivalence class. The main tool of our
analysis is the structure of the companion tableau of a ribbon
Littlewood-Richardson (LR) tableau detected by the descent set defined by the
composition whose parts are the ribbon row lengths.
