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Spring 2020, Math 621: Week 8 Problem Set Due: Friday, March 27Th, 2020 Applications to Enumerative Combinatorics

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Spring 2020, Math 621: Week 8 Problem Set Due: Friday, March 27Th, 2020 Applications to Enumerative Combinatorics Spring 2020, Math 621: Week 8 Problem Set Due: Friday, March 27th, 2020 Applications to Enumerative Combinatorics Discussion problems. The problems below should be worked on in groups, but will not be submitted for credit. Only submit the homework problems at the end of this document. The content included covers enough material to complete the assigned problems, but if you are interested in further reading, I suggest Chapter 1 of Combinatorial Reciprocity Theorems by Matthias Beck and Raman Sanyal (freely available from the first author's webpage). This week, we will be examining an application of polytopes and Ehrhart's theorem to com- binatorial objects called posets. First, the obligatory (and less useful) technical definition: a partially ordered set (or poset) is a set Π with a partial order that is reflexive, antisymmetric, and transitive (here, \partial" means that some elements p; q 2 Π are incomparable, in that neither p q nor q p). Having stated the technical definition, note that this is not the best way to think of a poset! For all of our posets, Π = fp1; : : : ; pdg will be finite, and the easiest way to specify/think about a finite poset is with its Hasse diagram, wherein each point is an element of Π, and two elements pi and pj lie \above" each other whenever pi pj. Below are several examples of Hasse diagrams, and the posets they define will be used frequently through the rest of this sheet. <latexit sha1_base64="hqQA8Kft+q9PtGnxfooYMs3F7/M=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0m0oMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0kPRr/XLFrbpzkFXi5aQCORr98ldvELM0QmmYoFp3PTcxfkaV4UzgtNRLNSaUjekQu5ZKGqH2s/mpU3JmlQEJY2VLGjJXf09kNNJ6EgW2M6JmpJe9mfif101NeO1nXCapQckWi8JUEBOT2d9kwBUyIyaWUKa4vZWwEVWUGZtOyYbgLb+8SloXVe+y6t7XKvWbPI4inMApnIMHV1CHO2hAExgM4Rle4c0Rzovz7nwsWgtOPnMMf+B8/gAEto2d</latexit> <latexit 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