Lattice paths and the antiautomorphism of the poset of descending plane partitions

Abstract

Mills et al. (J. Combin. Theory Ser. A 34 (1983) 340–359) defined the poset of descending plane partitions. As they showed, it has a unique antiautomorphism, denoted τ, which plays a central role in some of their conjectures about alternating sign matrices (where τ seems to translate into the reversal of the order of the columns). Although they describe τ explicitly, its combinatorial significance is not directly apparent. On the other hand, descending plane partitions are encoded readily in terms of lattice paths (using Gessel–Viennot methodology (Adv. in Math. 58 (1985) 300–321)). In this context, τ has a simple interpretation: it is Gessel–Viennot paths duality.