Discrete Mathematics & Theoretical Computer Science (Jan 2012)

Promotion and Rowmotion

  • Jessica Striker,
  • Nathan Williams

DOI
https://doi.org/10.46298/dmtcs.3038
Journal volume & issue
Vol. DMTCS Proceedings vol. AR,..., no. Proceedings

Abstract

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We present an equivariant bijection between two actions—promotion and rowmotion—on order ideals in certain posets. This bijection simultaneously generalizes a result of R. Stanley concerning promotion on the linear extensions of two disjoint chains and certain cases of recent work of D. Armstrong, C. Stump, and H. Thomas on noncrossing and nonnesting partitions. We apply this bijection to several classes of posets, obtaining equivariant bijections to various known objects under rotation. We extend the same idea to give an equivariant bijection between alternating sign matrices under rowmotion and under B. Wieland's gyration. Lastly, we define two actions with related orders on alternating sign matrices and totally symmetric self-complementary plane partitions.

Keywords