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Combinatorial properties of maps on finite posets

University of British Columbia

In this thesis, we make progress on the problem of enumerating tableaux on non-classical shapes by introducing a general family of P-partitions that we call periodic P-partitions. Such a family of P-partitions generalizes the parallelogramic shapes, which were analysed by L´opez, Mart´ınez, P´erez, P´erez, Basova, Sun, Tewari, and van Willigenburg, and certain truncated shifted shapes, where truncated shifted shapes were investigated by Adin, King, Roichman, and Panova. By introducing a separation property for posets and by proving a relationship between this property and P-partitions, we prove that periodic P-partitions can be enumerated with a homogeneous first-order matrix difference equation. Afterwards, we consider families of finite sets that we call shellable and that have been characterized by Chang and by Hirst and Hughes as being the families of sets that admit unique solutions to Hall’s marriage problem. By introducing constructions on families of sets that satisfy Hall’s Marriage Condition, and by using a combinatorial analogue of a shelling order, we prove that shellable families can be characterized by using a generalized notion of hook-lengths. Then, we introduce a natural generalization of standard skew tableaux and Edelman and Greene’s balanced tableaux, then prove an existence result about such a generalization using our characterization of shellable families.

Text

2020-07-16

Attribution 4.0 International

http://hdl.handle.net/2429/75185

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by/4.0/