Depths and Stanley depths of path ideals of spines. (English) Zbl 1328.05078

Summary: 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.


05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05E40 Combinatorial aspects of commutative algebra
13C14 Cohen-Macaulay modules
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
13A15 Ideals and multiplicative ideal theory in commutative rings
05C65 Hypergraphs
05C05 Trees
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