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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.
Keywords
Edge ideal, depth, path ideal, Cohen–Macaulay, monomial
ideal
Mathematical Subject Classification 2010
Primary: 05E40, 13C14, 13F55
Secondary: 13A15, 05C25, 05C65, 05C05
Milestones
Received: 2 October 2014
Revised: 22 December 2014
Accepted: 9 January 2015
Published: 17 December 2015
Communicated by Scott T. Chapman