# 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.
Language English p2.11 19 The Electronic Journal of Combinatorics 21 2 Published - 16 Apr 2014

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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title = "On the M{\"o}bius function of permutations with one descent",
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.",
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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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T1 - On the Möbius function of permutations with one descent

AU - Smith, Jason P.

PY - 2014/4/16

Y1 - 2014/4/16

N2 - 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.

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.

KW - math.CO

KW - 05A05

KW - Möbius function

KW - poset

KW - permutations

UR - http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i2p11

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JO - The Electronic Journal of Combinatorics

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JF - The Electronic Journal of Combinatorics

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