### Abstract

Language | English |
---|---|

Article number | p2.11 |

Number of pages | 19 |

Journal | The Electronic Journal of Combinatorics |

Volume | 21 |

Issue number | 2 |

Publication status | Published - 16 Apr 2014 |

### Fingerprint

### Keywords

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

### Cite this

*The Electronic Journal of Combinatorics*,

*21*(2), [p2.11].

}

*The Electronic Journal of Combinatorics*, vol. 21, no. 2, p2.11.

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

Research output: Contribution to journal › Article

TY - JOUR

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

M3 - Article

VL - 21

JO - The Electronic Journal of Combinatorics

T2 - The Electronic Journal of Combinatorics

JF - The Electronic Journal of Combinatorics

SN - 1077-8926

IS - 2

M1 - p2.11

ER -