Vol. 9, No. 1, 2016

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Depths and Stanley depths of path ideals of spines

Daniel Campos, Ryan Gunderson, Susan Morey, Chelsey Paulsen and Thomas Polstra

Vol. 9 (2016), No. 1, 155–170

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.

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

Communicated by Scott T. Chapman
Daniel Campos
525 West Mulberry Avenue
San Antonio, TX 78212
United States
Ryan Gunderson
Department of Mathematics
Duke University
Durham, NC 27708
United States
Susan Morey
Department of Mathematics
Texas State University
601 University Drive
San Marcos, TX 78666
United States
Chelsey Paulsen
East Chapel Hill High School
500 Weaver Dairy Road
Chapel Hill, NC 27514
United States
Thomas Polstra
Department of Mathematics
University of Missouri
202 Mathematical Sciences Building
Columbia, MO 65211
United States