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Full-Text Articles in Physical Sciences and Mathematics
Positivity Among P-Partition Generating Functions Of Partially Ordered Sets, Nate Lesnevich
Positivity Among P-Partition Generating Functions Of Partially Ordered Sets, Nate Lesnevich
Honors Theses
We find necessary and separate sufficient conditions for the difference between two labeled partially ordered set's (poset) partition generating functions to be positive in the fundamental basis. We define the notion of a jump sequence for a poset and show how different conditions on the jump sequences of two posets are necessary for those posets to have an order relation in the fundamental basis. Our sufficient conditions are of two types. First, we show how manipulating a poset's Hasse diagram produces a poset that is greater according to the fundamental basis. Secondly, we also provide tools to explain posets that …
Extensions Of The Morse-Hedlund Theorem, Eben Blaisdell
Extensions Of The Morse-Hedlund Theorem, Eben Blaisdell
Honors Theses
Bi-infinite words are sequences of characters that are infinite forwards and backwards; for example "...ababababab...". The Morse-Hedlund theorem says that a bi-infinite word f repeats itself, in at most n letters, if and only if the number of distinct subwords of length n is at most n. Using the example, "...ababababab...", there are 2 subwords of length 3, namely "aba" and "bab". Since 2 is less than 3, we must have that "...ababababab..." repeats itself after at most 3 letters. In fact it does repeat itself every two letters. …