an:07058690
Zbl 1417.06003
Striker, Jessica; Williams, Nathan
Promotion and rowmotion
EN
Proceedings of the 24th international conference on formal power series and algebraic combinatorics, FPSAC 2012, Nagoya, Japan, July 30--August 3, 2012. Nancy: The Association. Discrete Mathematics \& Theoretical Computer Science (DMTCS). Discrete Math. Theor. Comput. Sci., Proc., 271-283 (2012).
2012
a
06A07 05A19 05B20
poset; order ideal; noncrossing; promotion; equivariant; alternating sign matrices
Summary: We present an equivariant bijection between two actions -- promotion and rowmotion -- on order ideals in certain posets. This bijection simultaneously generalizes a result of \textit{R. P. Stanley} [Electron. J. Comb. 16, No. 2, Research Paper R9, 24 p. (2009; Zbl 1169.06002)] concerning promotion on the linear extensions of two disjoint chains and certain cases of recent work of [\textit{D. Armstrong, C. Stump} and \textit{H. Thomas}, ``A uniform bijection between noncrossing and nonnesting partitions'', Preprint, \url{arXiv:math.CO/1101.1277}] 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 \textit{B. Wieland}'s gyration [Electron. J. Comb. 7, No. 1, Research paper R37, 13 p. (2000; Zbl 0956.05015)]. Lastly, we define two actions with related orders on alternating sign matrices and totally symmetric self-complementary plane partitions.
For the entire collection see [Zbl 1257.05001].
Zbl 1169.06002; Zbl 0956.05015