Propp recently introduced regions in the hexagonal grid called benzels and stated several enumerative conjectures about the tilings of benzels using two types of prototiles called stones and bones. We resolve two of his conjectures and prove some additional results that he left tacit. In order to solve these problems, we first transfer benzels into the square grid. One of our primary tools, which we combine with several new ideas, is a bijection (rediscovered by Stanton and White and often attributed to them although it is considerably older) between \(k\)-ribbon tableaux of certain skew shapes and certain \(k\)-tuples of Young tableaux.

Mathematics Subject Classifications: 05B45, 05A15, 05A17

Keywords: Tiling, benzel, abacus bijection, core partition, domino, stone, bone

The rowmotion operator acting on the set of order ideals of a finite poset has been the focus of a significant amount of recent research. One of the major goals has been to exhibit homomesies: statistics that have the same average along every orbit of the action. We systematize a technique for proving that various statistics of interest are homomesic by writing these statistics as linear combinations of "toggleability statistics" (originally introduced by Striker) plus a constant. We show that this technique recaptures most of the known homomesies for the posets on which rowmotion has been most studied. We also show that the technique continues to work in modified contexts. For instance, this technique also yields homomesies for the piecewise-linear and birational extensions of rowmotion; furthermore, we introduce a \(q\)-analogue of rowmotion and show that the technique yields homomesies for "\(q\)-rowmotion" as well.

Mathematics Subject Classifications: 06A07, 05E18, 05A30, 52B05

Keywords: Homomesy, rowmotion, toggling, piecewise-linear & birational lift, \(q\)-analogue