On the Möbius function of permutations with one descent

Jason P. Smith

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

The set of all permutations, ordered by pattern containment, is a poset. We give a formula for the M\"obius function of intervals $[1,\pi]$ in this poset, for any permutation $\pi$ with at most one descent. We compute the M\"obius 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 M\"obius function is unbounded on the poset of all permutations. We show that the M\"obius 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 M\"obius function on some other intervals of permutations with at most one descent.
LanguageEnglish
Article numberp2.11
Number of pages19
JournalThe Electronic Journal of Combinatorics
Volume21
Issue number2
Publication statusPublished - 16 Apr 2014

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Möbius Function
M-function
Pi
Descent
Permutation
Consecutive
Poset
Interval
Monotone
Zero

Keywords

  • math.CO
  • 05A05
  • Möbius function
  • poset
  • permutations

Cite this

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On the Möbius function of permutations with one descent. / Smith, Jason P.

In: The Electronic Journal of Combinatorics, Vol. 21, No. 2, p2.11, 16.04.2014.

Research output: Contribution to journalArticle

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AB - The set of all permutations, ordered by pattern containment, is a poset. We give a formula for the M\"obius function of intervals $[1,\pi]$ in this poset, for any permutation $\pi$ with at most one descent. We compute the M\"obius 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 M\"obius function is unbounded on the poset of all permutations. We show that the M\"obius 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 M\"obius function on some other intervals of permutations with at most one descent.

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