| 1. |
- Abdallah, Nancy, et al.
(author)
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Combinatorial invariance of Kazhdan-Lusztig-Vogan polynomials for fixed point free involutions
- 2018
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In: Journal of Algebraic Combinatorics. - : SPRINGER. - 0925-9899 .- 1572-9192. ; 47:4, s. 543-560
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Journal article (peer-reviewed)abstract
- When acts on the flag variety of , the orbits are in bijection with fixed point free involutions in the symmetric group . In this case, the associated Kazhdan-Lusztig-Vogan polynomials can be indexed by pairs of fixed point free involutions , where denotes the Bruhat order on . We prove that these polynomials are combinatorial invariants in the sense that if is a poset isomorphism of upper intervals in the Bruhat order on fixed point free involutions, then for all v amp;gt;= u.
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| 2. |
- Ahmed, Chwas, et al.
(author)
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The graded Betti numbers of truncation of ideals in polynomial rings
- 2023
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In: Journal of Algebraic Combinatorics. - 0925-9899 .- 1572-9192. ; 57:4, s. 1303-1312
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Journal article (peer-reviewed)abstract
- Let R=K[x1,…,xn], a graded algebra S=R/I satisfies Nk,p if I is generated in degree k, and the graded minimal resolution is linear the first p steps, and the k-index of S is the largest p such that S satisfies Nk,p. Eisenbud and Goto have shown that for any graded ring R/I, then R/I≥k, where I≥k=I∩Mk and M=(x1,…,xn), has a k-linear resolution (satisfies Nk,p for all p) if k≫0. For a squarefree monomial ideal I, we are here interested in the ideal Ik which is the squarefree part of I≥k. The ideal I is, via Stanley–Reisner correspondence, associated to a simplicial complex ΔI. In this case, all Betti numbers of R/Ik for k>min{deg(u)∣u∈I}, which of course are a much finer invariant than the index, can be determined from the Betti diagram of R/I and the f-vector of ΔI. We compare our results with the corresponding statements for I≥k. (Here I is an arbitrary graded ideal.) In this case, we show that the Betti numbers of R/I≥k can be determined from the Betti numbers of R/I and the Hilbert series of R/I≥k.
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| 3. |
- Alexandersson, Per
(author)
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LLT polynomials, elementary symmetric functions and melting lollipops
- 2021
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In: Journal of Algebraic Combinatorics. - : Springer Science and Business Media LLC. - 0925-9899 .- 1572-9192. ; 53:2, s. 299-325
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Journal article (peer-reviewed)abstract
- We conjecture an explicit positive combinatorial formula for the expansion of unicellular LLT polynomials in the elementary symmetric basis. This is an analogue of the Shareshian-Wachs conjecture previously studied by Panova and the author in 2018. We show that the conjecture for unicellular LLT polynomials implies a similar formula for vertical-strip LLT polynomials. We prove positivity in the elementary symmetric basis for the class of graphs called melting lollipops previously considered by Huh, Nam and Yoo. This is done by proving a curious relationship between a generalization of charge and orientations of unit-interval graphs. We also provide short bijective proofs of Lee's three-term recurrences for unicellular LLT polynomials, and we show that these recurrences are enough to generate all unicellular LLT polynomials associated with abelian area sequences.
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| 4. |
- Amini, Nima
(author)
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Spectrahedrality of hyperbolicity cones of multivariate matching polynomials
- 2019
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In: Journal of Algebraic Combinatorics. - : SPRINGER. - 0925-9899 .- 1572-9192. ; 50:2, s. 165-190
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Journal article (peer-reviewed)abstract
- The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. We prove the conjecture for a multivariate generalization of the matching polynomial. This is further extended (albeit in a weaker sense) to a multivariate version of the independence polynomial for simplicial graphs. As an application, we give a new proof of the conjecture for elementary symmetric polynomials (originally due to Branden). Finally, we consider a hyperbolic convolution of determinant polynomials generalizing an identity of Godsil and Gutman.
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| 5. |
- Amini, Nima
(author)
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Spectrahedrality of hyperbolicity cones of multivariate matching polynomials
- 2018
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In: Journal of Algebraic Combinatorics. - : Springer. - 0925-9899 .- 1572-9192.
-
Journal article (peer-reviewed)abstract
- The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. We prove the conjecture for a multivariate generalization of the matching polynomial. This is further extended (albeit in a weaker sense) to a multivariate version of the independence polynomial for simplicial graphs. As an application we give a new proof of the conjecture for elementary symmetric polynomials (originally due to Brändén). Finally we consider a hyperbolic convolution of determinant polynomials generalizing an identity of Godsil and Gutman.
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| 6. |
- Assarf, Benjamin, et al.
(author)
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A bound for the splitting of smooth Fano polytopes with many vertices
- 2016
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In: Journal of Algebraic Combinatorics. - : Springer Science and Business Media LLC. - 0925-9899 .- 1572-9192. ; 43:1, s. 153-172
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Journal article (peer-reviewed)abstract
- The classification of toric Fano manifolds with large Picard number corresponds to the classification of smooth Fano polytopes with large number of vertices. A smooth Fano polytope is a polytope that contains the origin in its interior and is such that the vertex set of each facet forms a lattice basis. Casagrande showed that any smooth d-dimensional Fano polytope has at most 3d vertices. Smooth Fano polytopes in dimension d with at least vertices are completely known. The main result of this paper deals with the case of vertices for k fixed and d large. It implies that there is only a finite number of isomorphism classes of toric Fano d-folds X (for arbitrary d) with Picard number such that X is not a product of a lower-dimensional toric Fano manifold and the projective plane blown up in three torus-invariant points. This verifies the qualitative part of a conjecture in a recent paper by the first author, Joswig, and Paffenholz.
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| 7. |
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| 8. |
- Bousquet-Melou, Mireille, et al.
(author)
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On the independence complex of square grids
- 2008
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In: Journal of Algebraic Combinatorics. - : Springer Science and Business Media LLC. - 0925-9899 .- 1572-9192. ; 27:4, s. 423-450
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Journal article (peer-reviewed)abstract
- The enumeration of independent sets of regular graphs is of interest in statistical mechanics, as it corresponds to the solution of hard-particle models. In 2004, it was conjectured by Fendley et al., that for some rectangular grids, with toric boundary conditions, the alternating number of independent sets is extremely simple. More precisely, under a coprimality condition on the sides of the rectangle, the number of independent sets of even and odd cardinality always differ by 1. In physics terms, this means looking at the hard-particle model on these grids at activity -1. This conjecture was recently proved by Jonsson. Here we produce other families of grid graphs, with open or cylindric boundary conditions, for which similar properties hold without any size restriction: the number of independent sets of even and odd cardinality always differ by 0, +/- 1, or, in the cylindric case, by some power of 2. We show that these results reflect a stronger property of the independence complexes of our graphs. We determine the homotopy type of these complexes using Forman's discrete Morse theory. We find that these complexes are either contractible, or homotopic to a sphere, or, in the cylindric case, to a wedge of spheres. Finally, we use our enumerative results to determine the spectra of certain transfer matrices describing the hard-particle model on our graphs at activity -1. These results parallel certain conjectures of Fendley et al., proved by Jonsson in the toric case.
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| 9. |
- Bränden, Petter
(author)
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On operators on polynomials preserving real-rootedness and the Neggers-Stanley conjecture
- 2004
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In: Journal of Algebraic Combinatorics. - 0925-9899 .- 1572-9192. ; 20:2, s. 119-130
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Journal article (peer-reviewed)abstract
- We refine a technique used in a paper by Schur on real-rooted polynomials. This amounts to an extension of a theorem of Wagner on Hadamard products of Polya frequency sequences. We also apply our results to polynomials for which the Neggers-Stanley Conjecture is known to hold. More precisely, we settle interlacing properties for E-polynomials of series-parallel posets and column-strict labelled Ferrers posets.
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| 10. |
- Deligeorgaki, Danai
(author)
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Smallest graphs with given automorphism group
- 2022
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In: Journal of Algebraic Combinatorics. - : Springer Nature. - 0925-9899 .- 1572-9192.
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Journal article (peer-reviewed)abstract
- For a finite group G, denote by α(G) the minimum number of vertices of any graph having Aut() ∼= G. In this paper, we prove that α(G) ≤ |G|, with specifiedexceptions. The exceptions include four infinite families of groups, and 17 other smallgroups. Additionally, we compute α(G) for the groups G such that α(G) > |G| wherethe value α(G) was previously unknown.
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