## Higher Lawrence configurations

Venue: | J. Combin. Theory Ser. A |

Citations: | 29 - 1 self |

### BibTeX

@ARTICLE{Santos_higherlawrence,

author = {Francisco Santos and Bernd Sturmfels},

title = {Higher Lawrence configurations},

journal = {J. Combin. Theory Ser. A},

year = {},

pages = {151--164}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Any configuration of lattice vectors gives rise to a hierarchy of higher-dimensional configurations which generalize the Lawrence construction in geometric combinatorics. We prove finiteness results for the Markov bases, Graver bases and facet posets of these configurations, and we discuss applications to the statistical theory of log-linear models. 1.

### Citations

273 |
Ordering by divisibility in abstract algebras
- Higman
- 1952
(Show Context)
Citation Context ...d to be Noetherian if every non-empty subset of S has at least one, but at most finitely many minimal elements. Higman proved that if S is a Noetherian poset then � S is also Noetherian. In his paper =-=[8]-=-, he attributes this result to an earlier unpublished manuscript of Erdös and Rado. We apply this to the poset S = Z n , defined as in the introduction: a ≤ b ⇐⇒ for all i ∈ {1,... ,n}: 0 ≤ ai ≤ bi or... |

182 | Algebraic algorithms for sampling from conditional distributions
- Diaconis, Sturmfels
- 1998
(Show Context)
Citation Context ... the r-th Lawrence lifting of A. In this paper we study the Lawrence hierarchy A (2) , A (3) , A (4) ,... from the perspective of toric algebra, geometric combinatorics and applications to statistics =-=[4]-=-. The r-th Lawrence lifting A (r) is characterized as the configuration whose linear relations are r-tuples of linear relations on A that sum to zero. Indeed, the lattice of linear relations on A (r) ... |

116 |
The Analysis of Cross-Classified Categorical Data (Second Edition
- Fienberg
- 1980
(Show Context)
Citation Context ...linear models for m-dimensional contingency tables. Such a model is specified by a collection ∆ of subsets of {1,2,... ,m}. The standard notation for ∆, used in the books of Christensen [3], Fienberg =-=[5]-=- and other texts on cross-classified data, is a string of brackets each containing the elements of a subset in ∆. For instance, the four-cycle model for 4-dimensional tables is ∆ = � {1,2}, {2,3}, {3,... |

41 | Minimal basis for a connected Markov chain over 3 × 3 × K contingency tables with fixed two-dimensional marginals
- Aoki, Takemura
(Show Context)
Citation Context ... toric ideal I A (r) as in [10, §4], or, equivalently, to a minimal set of moves which connects any two nonnegative integer r × ntables that have the same column sums and the same A-weighted row sums =-=[1]-=-, [4], [6], [9]. We prove that the Markov bases stabilize for r ≫ 0. Theorem 1. For any configuration A = {a1,...,an} in Z d , there exists a constant m = m(A) such that any higher Lawrence lifting A ... |

30 | Gröbner bases and polyhedral geometry of reducible and cyclic models
- Ho¸sten, Sullivant
- 2002
(Show Context)
Citation Context ...A (r) as in [10, §4], or, equivalently, to a minimal set of moves which connects any two nonnegative integer r × ntables that have the same column sums and the same A-weighted row sums [1], [4], [6], =-=[9]-=-. We prove that the Markov bases stabilize for r ≫ 0. Theorem 1. For any configuration A = {a1,...,an} in Z d , there exists a constant m = m(A) such that any higher Lawrence lifting A (r) , for any r... |

7 |
Lectures on Polytopes, Springer Graduate Texts in Mathematics
- Ziegler
- 1994
(Show Context)
Citation Context ...sition translates into a decomposition of some multiple of u into tables with strictly smaller support. � Proof of Theorem 18: The configuration C of Lemma 19 has rank n−d. � Recall (e.g. from [2] or =-=[11]-=-) that the oriented matroid of A (r) is specified by the collection of all sign patterns of circuits of A (r) . Theorem 18 implies: Corollary 20. If c = c(A) < r then the oriented matroid of the highe... |

4 |
On the toric algebra of graphical models. Microsoft Research Report MSR-TR-2002-47
- Geiger, Meek, et al.
- 2002
(Show Context)
Citation Context ...al I A (r) as in [10, §4], or, equivalently, to a minimal set of moves which connects any two nonnegative integer r × ntables that have the same column sums and the same A-weighted row sums [1], [4], =-=[6]-=-, [9]. We prove that the Markov bases stabilize for r ≫ 0. Theorem 1. For any configuration A = {a1,...,an} in Z d , there exists a constant m = m(A) such that any higher Lawrence lifting A (r) , for ... |

2 |
Log-Linear Models. Springer Texts in Statistics
- Christensen
- 1990
(Show Context)
Citation Context ...erarchical loglinear models for m-dimensional contingency tables. Such a model is specified by a collection ∆ of subsets of {1,2,... ,m}. The standard notation for ∆, used in the books of Christensen =-=[3]-=-, Fienberg [5] and other texts on cross-classified data, is a string of brackets each containing the elements of a subset in ∆. For instance, the four-cycle model for 4-dimensional tables is ∆ = � {1,... |

2 |
4ti2: Computation of Hilbert bases, Graver bases, toric Gröbner bases, and more. Software freely available at http://www.4ti2.de
- Hemmecke
(Show Context)
Citation Context ...t has � � � � 4 4 240 + · 87 + · 5 = 558 3 2 elements in total. We similarly compute the Graver bases for the higher Lawrence liftings A (5) , A (6) , ..., for instance, using Hemmecke’s program 4ti2 =-=[7]-=-. The Graver basis of A (6) contains the following table of type 6: (5) (120 tables, degree 15, type 6) ⎛ −2 ⎜ ⎜−2 ⎜ 1 ⎜ 1 ⎝ 1 3 3 −2 −2 −2 0 0 1 1 1 ⎞ −1 −1 ⎟ 0 ⎟ 0 ⎟ 0 ⎠ 1 0 −3 2 The Graver basis el... |

2 |
On the toric algebra of graphical models, submitted to the Annals of Statistics, available as Microsoft Research Preprint at http://research.microsoft.com/scripts/pubs/view.asp?TR
- Geiger, Meek, et al.
(Show Context)
Citation Context ...al I A (r) as in [10, §4], or, equivalently, to a minimal set of moves which connects any two nonnegative integer r × ntables that have the same column sums and the same A-weighted row sums [1], [4], =-=[6]-=-, [9]. We prove that the Markov bases stabilize for r ≫ 0. Theorem 1. For any configuration A = {a1,...,an} in Z d , there exists a constant m = m(A) such that every higher Lawrence lifting A (r) has ... |

1 |
MLP: A program for computation of minimal points in lattices (Graver and Hilbert bases), freely available at http://www.testsets.de
- Hemmecke
(Show Context)
Citation Context ...it has ( ) ( ) 4 4 240 + · 87 + · 5 = 558 3 2 elements in total. We similarly compute the Graver bases for the higher Lawrence liftings A (5) , A (6) , ..., for instance, using Hemmecke’s program MLP =-=[7]-=-. The Graver basis of A (6) contains the following table of type 6: ⎛ ⎞ −2 3 0 −1 −2 3 0 −1 (5) (120 tables, degree 15, type 6) 1 −2 1 0 ⎜ 1 −2 1 0 ⎟ ⎝ 1 −2 1 0 ⎠ 1 0 −3 2 The Graver basis element (5)... |