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Counting words avoiding patterns of length three with generating functions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2015 (English)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
Abstract [en]

The set of 123-avoiding words with exactly r occurrences of each letter was recently enumerated by N. Shar and D. Zeilberger. This paper enumerates more complicated sets of pattern avoiding words, such as those allowing 1, 2, ... or r occurrences of each letter, or those for which the numbers of occurrences of each individual letter follow a repeating sequence. The results are also generalized to apply to all patterns of length three with distinct letters. The generating functions enumerating the words are shown to be algebraic, for all investigated sets of words. A notable number of coefficients for the relevant generating functions have been found and the first few confirmed by independent methods. The asymptotic behaviour of these coefficients has been established as exponential. The employed strategy involves partitioning words into subwords which allow for the construction of equations relating the generating functions.

Abstract [sv]

Mängden av 123-undvikande ord med exakt r förekomster av varje bokstav enumererades nyligen av N. Shar och D. Zeilberger. Detta arbete enumererar mer komplicerade mängder av mönsterundvikande ord, såsom de som tillåter 1,2, ... eller r förekomster av varje bokstav, eller de för vilka antalet förekomster av varje enskild bokstav följer en upprepande sekvens. Resultaten generaliseras även till att gälla alla mönster av längd tre med distinkta bokstäver. De genererande funktionerna som enumererar orden bevisas vara algebraiska, för alla undersökta ordmängder. Ett ansenligt antal koefficienter för de relevanta genererande funktionerna har beräknats och de första av dessa har bekräftats med oberoende metoder. Dessa koefficienters asymptotiska beteende har visats vara exponentiellt. Metoden som utnyttjas involverar en uppdelning i delord som tillåter konstruerandet av ekvationer som relaterar de genererande funktionerna.

Place, publisher, year, edition, pages
2015. , 48 p.
National Category
URN: urn:nbn:se:kth:diva-168019OAI: diva2:813933
Available from: 2015-05-25 Created: 2015-05-25 Last updated: 2015-05-27Bibliographically approved

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