#### DMCA

## LATTICE POINT GENERATING FUNCTIONS AND SYMMETRIC CONES

### Citations

689 |
Reflection Groups and Coxeter Groups
- Humphreys
- 1990
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Citation Context ...oups, Coxeter groups, and descents. In the following, we will consider finite reflection groups (i.e., finite subgroups of O(V) for some Euclidean space V that are generated by reflections—see, e.g., =-=[10]-=- for background) that act on the underlying Euclidean space in a restricted fashion. Namely, a finite reflection group W ⊂ O(V) acting on a Euclidean vector space V is called almost irreducible if V d... |

282 |
Groupes et algebres de Lie: Chapitres 4, 5 et 6.
- Bourbaki
- 1968
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Citation Context ...d in [3]. We include this case here to show how the result can be derived from Theorem 2.8. Let Sn denote the group of permutations of the set {1,...,n}. For π ∈ Sn, we define the descent set of π as =-=(6)-=- D(π) := {j ∈ {1,...,n−1} | π(j) > π(j +1)}. This is the standard definition used in the literature on permutations. The group Sn acts on R n by permutation of the components. For π ∈ Sn, let σπ ∈ O(R... |

278 |
Combinatorics of Coxeter groups
- Björner, Brenti
- 2005
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Citation Context ...orresponding relation. Such generators are called simple generators. For each Coxeter group considered, we will suppose that simple generators have been fixed once and for all. We refer the reader to =-=[5]-=- or [10] for further information about Coxeter groups and their relation to reflection groups. The length l(σ) of an element σ ∈ W of a Coxeter group W is the smallest integer such that there is a dec... |

143 |
Ordered Structures and Partitions,
- Stanley
- 1972
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Citation Context ...t sjσ−1x ∈ C+. Hence σ−1x ∈ Hj. Since l(σsj) < l(σ) we have j ∈ Dr(σ). Hence σ−1x ∈ CDr(σ), and so x ∈ σCDr(σ), a contradiction. □ Our proposition above is reminiscent of the theory of P-partitions =-=[12, 13]-=-. For a givenfiniteposetP, onecanproduceaconeofP-partitions. Thestandardapproach to studying a P-partition cone, originating in the work of Stanley referenced above, is to recognize that each such con... |

120 | Computing the continuous discretely: Integerpoint enumeration in polyhedra. Springer Undergraduate Texts in Mathematics,
- Beck, Robins
- 2007
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Citation Context ...ors of W. Let l denote the corresponding length function. Let x ∈ F ∩V1 and σ ∈ W. Let H be a facet hyperplane of σF, such that l(sσ) > l(σ) for the reflection s at H. We claim that in this situation =-=(4)-=- (sσx,x ′ 0 ) ≤ (σx,x′ 0 ). Indeed, consider the decomposition V1 = (H ∩V1)⊕H ⊥ . According to this decomposition, write x ′ 0 = v0 + w0 and σx = v1 + w1; then sσx = v1 − w1. We have (σx,x ′ 0) = (v1,... |

66 |
q-Eulerian polynomials arising from Coxeter groups, preprint
- Brenti
- 1993
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Citation Context ...an polynomials. Intheremainderofsection 4, we provide applications of Proposition 4.2 with connections to permutation statistics and Ehrhart theory. Our first application is well known, going back to =-=[8]-=- and [14]; the polyhedral perspective of the following identity was first established in [14], also using Ehrhart theory. Corollary 4.3 ([8], [14]). The hyperoctahedral Eulerian polynomials are given ... |

29 |
with a foreword by Gian-Carlo Rota, corrected reprint of the 1986 original
- Vol
- 1997
(Show Context)
Citation Context ...t sjσ−1x ∈ C+. Hence σ−1x ∈ Hj. Since l(σsj) < l(σ) we have j ∈ Dr(σ). Hence σ−1x ∈ CDr(σ), and so x ∈ σCDr(σ), a contradiction. □ Our proposition above is reminiscent of the theory of P-partitions =-=[12, 13]-=-. For a givenfiniteposetP, onecanproduceaconeofP-partitions. Thestandardapproach to studying a P-partition cone, originating in the work of Stanley referenced above, is to recognize that each such con... |

28 | On the descent numbers and major indices for the hyperoctahedral group,
- Chow, Gessel
- 2007
(Show Context)
Citation Context ...SAVAGE for (π,ε) ∈ Bn−1. Similarly, the major index is maj(π,ε) := ∑ (j −1) and the comajor index is j∈D(π,ε) comaj(π,ε) := ∑ j∈D(π,ε) (n−j) for (π,ε) ∈ Bn−1. It follows that we have the relationship =-=(9)-=- comaj(π,ε) = (n−1)des(π,ε)−maj(π,ε). 4.1. The multivariate generating function. Inthissituation,Corollary 2.9specializes as follows. Proposition 4.2. Fix integers 0 ≤ a1 ≤ ··· ≤ an−1 = 0. Let C := {... |

18 |
Integer points in polyhedra
- Barvinok
- 2008
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Citation Context ... ∑ (π,ε)∈Bn−1 t des(π,ε) . (2k +1) n−1 t k . t des(π,ε) = ∑ k≥0 (2k +1) n−1 t k 4.3. The distribution of the comajor index. For k ∈ N and a variable t, let [k]t := 1+t+t 2 +···+t k−1 and [k]t! := [1]t=-=[2]-=-t···[k]t. We show here how to derive the distribution of the comajor index. This is likely well-known, as for example it follows from (10) below, but we could not find an explicit statement in the lit... |

11 | MacMahon’s partition analysis. VII. Constrained compositions. In q-series with applications to combinatorics, number theory, and physics
- Andrews, Paule, et al.
- 2000
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Citation Context ...xpressions in type B, including identities involving permutation statistics and lecture hall partitions. 1. Introduction Motivated by the “constrained compositions” introduced by Andrews–Paule– Riese =-=[1]-=-, Beck–Gessel–Lee–Savage [3] enumerated symmetrically constrained compositions, i.e., compositions of an integer M into n nonnegative parts M = λ1 +λ2 +···+λn, where the sequence λ := (λ1,λ2,...,λn) s... |

5 | Lecture hall partitions and the wreath products Ck ≀Sn, preprint
- Pensyl, Savage
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Citation Context ...t Ln = {λ ∈ Z n | 0 ≤ λ1 1 ≤ λ2 2 ≤ ··· ≤ λn n }. The following relationship between statistics on lecture hall partitions and statistics on signed permutations follows from work of Pensyl and Savage =-=[11]-=-. Define the lecture hall polytope Pn,2 by Pn,2 = { λ ∈ R n ∣ 0 ≤ λ1 2 ≤ λ2 4 } λn ≤ ··· ≤ ≤ 1 . 2n For λ ∈ Ln, (π,ε) ∈ Bn, and ⌈x⌉ = inf([x,∞)∩Z), set n∑ ⌈ ⌉ n∑ ⌈ ⌉ λi λi stat1(λ) = , stat2(λ) = 2i ,... |

4 | Symmetrically constrained compositions
- Beck, Gessel, et al.
(Show Context)
Citation Context ...ing identities involving permutation statistics and lecture hall partitions. 1. Introduction Motivated by the “constrained compositions” introduced by Andrews–Paule– Riese [1], Beck–Gessel–Lee–Savage =-=[3]-=- enumerated symmetrically constrained compositions, i.e., compositions of an integer M into n nonnegative parts M = λ1 +λ2 +···+λn, where the sequence λ := (λ1,λ2,...,λn) satisfies the symmetric syste... |

4 |
Permutation statistics of indexed permutations, European
- Steingrimsson
- 1994
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Citation Context ...omials. Intheremainderofsection 4, we provide applications of Proposition 4.2 with connections to permutation statistics and Ehrhart theory. Our first application is well known, going back to [8] and =-=[14]-=-; the polyhedral perspective of the following identity was first established in [14], also using Ehrhart theory. Corollary 4.3 ([8], [14]). The hyperoctahedral Eulerian polynomials are given by ∑ ∞∑ (... |

1 |
Lecture hall partitions
- MireilleBousquet-Mélou, KimmoEriksson
- 1997
(Show Context)
Citation Context ... the previous subsections to allow ai to be a linear function of i, then fC(t) can be expressed in terms of lecture hall partitions. Lecture hall partitions, introduced by Bousquet-Mélou and Eriksson =-=[7]-=-, are elements of the set Ln = {λ ∈ Z n | 0 ≤ λ1 1 ≤ λ2 2 ≤ ··· ≤ λn n }. The following relationship between statistics on lecture hall partitions and statistics on signed permutations follows from wo... |