### Abstract

Let ℒ=(X,⪯) be a lattice. For P⊆X we say that P is t-intersecting if rank(x∧y)≥t for all x,y∈P. The seminal theorem of Erdős, Ko and Rado describes the maximum intersecting P in the lattice of subsets of a finite set with the additional condition that P is contained within a level of the lattice. The Erdős–Ko–Rado theorem has been extensively studied and generalized to other objects and lattices. In this paper, we focus on intersecting families of permutations as defined with respect to the weak Bruhat lattice. In this setting, we prove analogs of certain extremal results on intersecting set systems. In particular we give a characterization of the maximum intersecting families of permutations in the Bruhat lattice. We also characterize the maximum intersecting families of permutations within the rth level of the Bruhat lattice of permutations of size n, provided that n is large relative to r.

Original language | English (US) |
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Journal | Discrete Applied Mathematics |

DOIs | |

State | Accepted/In press - Jan 1 2019 |

### Fingerprint

### Keywords

- Bruhat lattice
- Erdős–Ko–Rado theorem
- Intersecting permutations

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete Applied Mathematics*. https://doi.org/10.1016/j.dam.2018.12.019

**Erdős–Ko–Rado theorems on the weak Bruhat lattice.** / Fishel, Susanna; Hurlbert, Glenn; Kamat, Vikram; Meagher, Karen.

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*. https://doi.org/10.1016/j.dam.2018.12.019

}

TY - JOUR

T1 - Erdős–Ko–Rado theorems on the weak Bruhat lattice

AU - Fishel, Susanna

AU - Hurlbert, Glenn

AU - Kamat, Vikram

AU - Meagher, Karen

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Let ℒ=(X,⪯) be a lattice. For P⊆X we say that P is t-intersecting if rank(x∧y)≥t for all x,y∈P. The seminal theorem of Erdős, Ko and Rado describes the maximum intersecting P in the lattice of subsets of a finite set with the additional condition that P is contained within a level of the lattice. The Erdős–Ko–Rado theorem has been extensively studied and generalized to other objects and lattices. In this paper, we focus on intersecting families of permutations as defined with respect to the weak Bruhat lattice. In this setting, we prove analogs of certain extremal results on intersecting set systems. In particular we give a characterization of the maximum intersecting families of permutations in the Bruhat lattice. We also characterize the maximum intersecting families of permutations within the rth level of the Bruhat lattice of permutations of size n, provided that n is large relative to r.

AB - Let ℒ=(X,⪯) be a lattice. For P⊆X we say that P is t-intersecting if rank(x∧y)≥t for all x,y∈P. The seminal theorem of Erdős, Ko and Rado describes the maximum intersecting P in the lattice of subsets of a finite set with the additional condition that P is contained within a level of the lattice. The Erdős–Ko–Rado theorem has been extensively studied and generalized to other objects and lattices. In this paper, we focus on intersecting families of permutations as defined with respect to the weak Bruhat lattice. In this setting, we prove analogs of certain extremal results on intersecting set systems. In particular we give a characterization of the maximum intersecting families of permutations in the Bruhat lattice. We also characterize the maximum intersecting families of permutations within the rth level of the Bruhat lattice of permutations of size n, provided that n is large relative to r.

KW - Bruhat lattice

KW - Erdős–Ko–Rado theorem

KW - Intersecting permutations

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UR - http://www.scopus.com/inward/citedby.url?scp=85059849926&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2018.12.019

DO - 10.1016/j.dam.2018.12.019

M3 - Article

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -