The M\"obius function of generalized subword order
Abstract
Let P be a poset and let P* be the set of all finite length words over P. Generalized subword order is the partial order on P* obtained by letting u \leq w if and only if there is a subword u' of w having the same length as u such that each element of u is less than or equal to the corresponding element of u' in the partial order on P. Classical subword order arises when P is an antichain, while letting P be a chain gives an order on compositions. For any finite poset P, we give a simple formula for the Mobius function of P* in terms of the Mobius function of P. This permits us to rederive in a easy and uniform manner previous results of Bjorner, Sagan and Vatter, and Tomie. We are also able to determine the homotopy type of all intervals in P* for any finite P of rank at most 1.
Cite
@article{arxiv.1107.5070,
title = {The M\"obius function of generalized subword order},
author = {Peter R. W. McNamara and Bruce E. Sagan},
journal= {arXiv preprint arXiv:1107.5070},
year = {2012}
}
Comments
29 pages, 4 figures. Incorporates referees' suggestions; to appear in Advances in Mathematics