Journal of Integer Sequences, Vol. 14 (2011), Article 11.4.2

## Generating Functions for Wilf Equivalence Under Generalized Factor Order

### Thomas Langley Department of Mathematics Rose-Hulman Institute of Technology Terre Haute, IN 47803 USA Jeffrey Liese Department of Mathematics California Polytechnic State University San Luis Obispo, CA 93407-0403 USA Jeffrey Remmel Department of Mathematics University of California, San Diego La Jolla, CA 92093-0112 USA

Abstract:

Kitaev, Liese, Remmel, and Sagan recently defined generalized factor order on words comprised of letters from a partially ordered set by setting if there is a contiguous subword of of the same length as such that the -th character of is greater than or equal to the -th character of for all . This subword is called an embedding of into . For the case where is the positive integers with the usual ordering, they defined the weight of a word to be wt, and the corresponding weight generating function    wt. They then defined two words and to be Wilf equivalent, denoted , if and only if . They also defined the related generating function    wt where is the set of all words such that the only embedding of into is a suffix of , and showed that if and only if . We continue this study by giving an explicit formula for if factors into a weakly increasing word followed by a weakly decreasing word. We use this formula as an aid to classify Wilf equivalence for all words of length 3. We also show that coefficients of related generating functions are well-known sequences in several special cases. Finally, we discuss a conjecture that if then and must be rearrangements, and the stronger conjecture that there also must be a weight-preserving bijection on words over the positive integers such that is a rearrangement of for all , and embeds if and only if embeds .

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(Concerned with sequences A000045 A000071 A000073 A000078 A000124 A000126 A000292 A001591 A001949 A007800 A008466 A008937 A014162 A050231 A050232 A050233 A107066 A145112 A145113 A172119.)

Received May 24 2010; revised version received March 16 2011. Published in Journal of Integer Sequences, March 26 2011.