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) |
|---|---|
| Journal | Discrete Applied Mathematics |
| DOIs | |
| State | Accepted/In press - Jan 1 2019 |
| Externally published | Yes |
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Keywords
- Bruhat lattice
- Erdős–Ko–Rado theorem
- Intersecting permutations
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Applied Mathematics
Cite this
Erdős–Ko–Rado theorems on the weak Bruhat lattice. / Fishel, Susanna; Hurlbert, Glenn; Kamat, Vikram; Meagher, Karen.
In: Discrete Applied Mathematics, 01.01.2019.Research output: Contribution to journal › Article
}
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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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 -