Depths and Stanley depths of path ideals of spines

Abstract

For a special class of trees, namely trees that are themselves a path, a precise formula is given for the depth of an ideal generated by all (undirected) paths of a fixed length. The dimension of these ideals is also computed, which is used to classify which such ideals are Cohen–Macaulay. The techniques of the proofs are shown to extend to provide a lower bound on the Stanley depth of these ideals. Combining these results gives a new class of ideals for which the Stanley conjecture holds.