On the Möbius function of permutations with one descent
The electronic journal of combinatorics, Tome 21 (2014) no. 2
The set of all permutations, ordered by pattern containment, is a poset. We give a formula for the Möbius function of intervals $[1,\pi]$ in this poset, for any permutation $\pi$ with at most one descent. We compute the Möbius function as a function of the number and positions of pairs of consecutive letters in $\pi$ that are consecutive in value. As a result of this we show that the Möbius function is unbounded on the poset of all permutations. We show that the Möbius function is zero on any interval $[1,\pi]$ where $\pi$ has a triple of consecutive letters whose values are consecutive and monotone. We also conjecture values of the Möbius function on some other intervals of permutations with at most one descent.
DOI :
10.37236/3559
Classification :
05A05, 06A07, 11N99
Mots-clés : Möbius function
Mots-clés : Möbius function
Affiliations des auteurs :
Jason P. Smith 1
@article{10_37236_3559,
author = {Jason P. Smith},
title = {On the {M\"obius} function of permutations with one descent},
journal = {The electronic journal of combinatorics},
year = {2014},
volume = {21},
number = {2},
doi = {10.37236/3559},
zbl = {1300.05019},
url = {http://geodesic.mathdoc.fr/articles/10.37236/3559/}
}
Jason P. Smith. On the Möbius function of permutations with one descent. The electronic journal of combinatorics, Tome 21 (2014) no. 2. doi: 10.37236/3559
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