Journal of Combinatorics

Volume 5 (2014)

Number 2

The $q = -1$ phenomenon via homology concentration

Pages: 167 – 194

DOI: https://dx.doi.org/10.4310/JOC.2014.v5.n2.a2

Authors

P. Hersh (Department of Mathematics, North Carolina State University, Raleigh, N.C., U.S.A.)

J. Shareshian (Department of Mathematics, Washington University, St. Louis, Missouri, U.S.A.)

D. Stanton (School of Mathematics, University of Minnesota, Minneapolis, Minn., U.S.A.)

Abstract

We introduce a homological approach to exhibiting instances of Stembridge’s $q = -1$ phenomenon. This approach is shown to explain two important instances of the phenomenon, namely that of partitions whose Ferrers diagrams fit in a rectangle of fixed size and that of plane partitions fitting in a box of fixed size. A more general framework of invariant and coinvariant complexes with coefficients taken mod 2 is developed, and as a part of this story an analogous homological result for necklaces is conjectured.

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Published 20 August 2014