| 1. |
- Alexandersson, Per, et al.
(författare)
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Properties of Non-symmetric Macdonald Polynomials at q=1 and q=0
- 2019
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Ingår i: Annals of Combinatorics. - : Springer Publishing Company. - 0218-0006 .- 0219-3094. ; 23:2, s. 219-239
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Tidskriftsartikel (refereegranskat)abstract
- We examine the non-symmetric Macdonald polynomials E at q=1, as well as the more general permuted-basement Macdonald polynomials. When q=1, we show that E(x;1,t) is symmetric and independent of t whenever is a partition. Furthermore, we show that, in general , this expression factors into a symmetric and a non-symmetric part, where the symmetric part is independent of t, and the non-symmetric part only depends on x, t, and the relative order of the entries in . We also examine the case q=0, which gives rise to the so-called permuted-basement t-atoms. We prove expansion properties of these t-atoms, and, as a corollary, prove that Demazure characters (key polynomials) expand positively into permuted-basement atoms. This complements the result that permuted-basement atoms are atom-positive. Finally, we show that the product of a permuted-basement atom and a Schur polynomial is again positive in the same permuted-basement atom basis. Haglund, Luoto, Mason, and van Willigenburg previously proved this property for the identity basement and the reverse identity basement, so our result can be seen as an interpolation (in the Bruhat order) between these two results. The common theme in this project is the application of basement-permuting operators as well as combinatorics on fillings, by applying results in a previous article by Per Alexandersson.
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| 2. |
- Backelin, Jörgen, et al.
(författare)
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Parity splits by triple point distances in X-trees
- 2006
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Ingår i: Annals of Combinatorics. - : Springer Science and Business Media LLC. - 0218-0006 .- 0219-3094. ; 10:1, s. 1-18
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Tidskriftsartikel (refereegranskat)abstract
- At the conference Dress defined parity split maps by triple point distance and asked for a characterisation of such maps coming from binary phylogenetic X-trees. This article gives an answer to that question. The characterisation for X-trees can be easily described as follows: If all restrictions of a split map to sets of five or fewer elements is a parity split map for an X-tree, then so is the entire map. To ensure that the parity split map comes from an X-tree which is binary and phylogenetic, we add two more technical conditions also based on studying at most five points at a time.
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| 3. |
- Beck, Matthias, et al.
(författare)
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Smooth Centrally Symmetric Polytopes in Dimension 3 are IDP
- 2019
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Ingår i: Annals of Combinatorics. - : Springer Publishing Company. - 0218-0006 .- 0219-3094. ; 23:2, s. 255-262
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Tidskriftsartikel (refereegranskat)abstract
- In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Oda's conjecture for centrally symmetric 3-dimensional polytopes, by showing they are covered by lattice parallelepipeds and unimodular simplices.
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| 4. |
- Björner, Anders, et al.
(författare)
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A note on blockers in posets
- 2004
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Ingår i: Annals of Combinatorics. - : Springer Science and Business Media LLC. - 0218-0006 .- 0219-3094. ; 8:2, s. 123-131
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Tidskriftsartikel (refereegranskat)abstract
- The blocker A* of an antichain A in a finite poset P is the set of elements minimal with the property of having with each member of A a common predecessor. The following is done: (1) The posets P for which A** = A for all antichains are characterized.(2) The blocker A* of a symmetric antichain in the partition lattice is characterized.(3) Connections with the question of finding minimal size blocking sets for certain set families are discussed.
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| 5. |
- Eriksen, Niklas, 1974-, et al.
(författare)
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Expected reflection distance in G(r, 1, n) after a fixed number of reflections
- 2005
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Ingår i: Annals of Combinatorics. - Basel, Switzerland : Springer Science and Business Media LLC. - 0218-0006 .- 0219-3094. ; 9:1, s. 21-33
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Tidskriftsartikel (refereegranskat)abstract
- Extending to r > 1 a formula of the authors, we compute the expected reflection distance of a product of t random reflections in the complex reflection group G (r, 1, n). The result relies on an explicit decomposition of the reflection distance function into irreducible G (r, 1, n) characters and on the eigenvalues of certain adjacency matrices.
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| 6. |
- Eriksson, Henrik, et al.
(författare)
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Dense packing of patterns in a permutation
- 2007
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Ingår i: Annals of Combinatorics. - : Springer Science and Business Media LLC. - 0218-0006 .- 0219-3094. ; 11:3-4, s. 459-470
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Tidskriftsartikel (refereegranskat)abstract
- We study the length L-k of the shortest permutation containing all patterns of length k. We establish the bounds e(-2)k(2) < L-k <= (2/3 + o(1))k(2). We also prove that as k there are permutations of length (1/4+o(1))k(2) containing almost all patterns of length k.
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| 7. |
- Ferroni, Luis, et al.
(författare)
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The Merino–Welsh Conjecture for Split Matroids
- 2023
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Ingår i: Annals of Combinatorics. - : Springer Nature. - 0218-0006 .- 0219-3094. ; 27:3, s. 737-748
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Tidskriftsartikel (refereegranskat)abstract
- In 1999, Merino and Welsh conjectured that evaluations of the Tutte polynomial of a graph satisfy an inequality. In this short article, we show that the conjecture generalized to matroids holds for the large class of all split matroids by exploiting the structure of their lattice of cyclic flats. This class of matroids strictly contains all paving and copaving matroids.
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| 8. |
- Incitti, Federico
(författare)
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Permutation diagrams, fixed points and Kazhdan-Lusztig R-polynomials
- 2006
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Ingår i: Annals of Combinatorics. - : Springer Science and Business Media LLC. - 0218-0006 .- 0219-3094. ; 10:3, s. 369-387
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we give an algorithm for computing the Kazhdan-Lusztig R-polynomials in the symmetric group. The algorithm is described in terms of permutation diagrams. In particular we focus on how the computation of the polynomial is affected by certain fixed points. As a consequence of our methods, we obtain explicit formulas for the R-polynomials associated with some general classes of intervals, generalizing results of Brenti and Pagliacci.
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| 9. |
- Johansson, Robert, et al.
(författare)
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Pattern avoidance in alternating sign matrices
- 2007
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Ingår i: Annals of Combinatorics. - : Springer Science and Business Media LLC. - 0218-0006 .- 0219-3094. ; 11:04-mar, s. 471-480
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Tidskriftsartikel (refereegranskat)abstract
- We generalize the definition of a pattern from permutations to alternating sign matrices. The number of alternating sign matrices avoiding 132 is proved to be counted by the large Schroder numbers, 1, 2, 6, 22, 90, 394,.... We give a bijection between 132-avoiding alternating sign matrices and Schroder paths, which gives a refined enumeration. We also show that the 132-, 123-avoiding alternating sign matrices are counted by every second Fibonacci number.
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| 10. |
- Jonsson, Jakob
(författare)
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On the 3-Torsion Part of the Homology of the Chessboard Complex
- 2010
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Ingår i: Annals of Combinatorics. - : Springer Science and Business Media LLC. - 0218-0006 .- 0219-3094. ; 14:4, s. 487-505
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Tidskriftsartikel (refereegranskat)abstract
- Let 1 (d) (M-m,M-n; Z) not equal 0. Second, for each k >= 0, we show that there is a polynomial f(k)(a, b) of degree 3k such that the dimension of (H) over tilde (k+a+2b-2) (M-k+a+3b-1,M- k+2a+3b-1; Z(3)), viewed as a vector space over Z(3), is at most f(k)(a, b) for all a >= 0 and b >= k+ 2. Third, we give a computer- free proof that (H) over tilde (2) (M-5,M-5; Z) congruent to Z(3). Several proofs are based on a new long exact sequence relating the homology of a certain subcomplex of M-m,M-n to the homology of M-m-2,M-n-1 and M-m-2,M-n-3.
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