On the Möbius Function of Permutations With One Descent
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by
Jason P Smith
2014
Abstract
The set of all permutations, ordered by pattern containment, is a poset. We
give a formula for the Möbius function of intervals [1,π] in this poset,
for any permutation π with at most one descent. We compute the Möbius
function as a function of the number and positions of pairs of consecutive
letters in π 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,π] where π
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.
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