| 1. |
- Björner, Anders
(författare)
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Nerves, fibers and homotopy groups
- 2003
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Ingår i: Journal of combinatorial theory. Series A (Print). - : Elsevier BV. - 0097-3165 .- 1096-0899. ; 102:1, s. 88-93
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Tidskriftsartikel (refereegranskat)abstract
- Two theorems are proved. One concerns coverings of a simplicial complex Delta by subcomplexes. It is shown that if every t-wise intersection of these subcomplexes is (k - t + 1)-connected, then for jless than or equal tok there are isomorphisms pi(j)(Delta) congruent to pi(j)(N) of homotopy groups of Delta and of the nerve X of the covering. The other concerns poset maps f : P --> Q. It is shown that if all fibers f(-1)(Q(less than or equal toq)) are k-connected, then f induces isomorphisms of homotopy groups pi(j)(P) congruent to pi(j)(Q), for all jless than or equal tok.
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| 2. |
- Hultman, Axel
(författare)
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Bruhat intervals of length 4 in Weyl groups
- 2003
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Ingår i: Journal of combinatorial theory. Series A (Print). - : Elsevier BV. - 0097-3165 .- 1096-0899. ; 102:1, s. 163-178
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Tidskriftsartikel (refereegranskat)abstract
- We determine all isomorphism classes of intervals of length 4 in the Bruhat order on the Weyl groups A(4), B-4, D-4 and F-4. It turns out that there are 24 of them (some of which are dual to each other). Work of Dyer allows us to conclude that these are the only intervals of length 4 that can occur in the Bruhat order on any Weyl group. We also determine the intervals that arise already in the smaller classes of simply laced Weyl groups and symmetric groups. Our method combines theoretical arguments and computer calculations. We also present an independent, completely computerized, approach.
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| 3. |
- Alexandersson, Per, 1987-
(författare)
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Stretched skew Schur polynomials are recurrent
- 2014
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Ingår i: Journal of combinatorial theory. Series A (Print). - : Elsevier BV. - 0097-3165 .- 1096-0899. ; 122, s. 1-8
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Tidskriftsartikel (refereegranskat)abstract
- We show that sequences of skew Schur polynomials obtained from stretched semi-standard Young tableauxsatisfy a linear recurrence, which we give explicitly.Using this, we apply this to finding certain asymptotic behavior of these Schur polynomials and present conjectures on minimal recurrences for stretched Schur polynomials.
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| 4. |
- Andrews, George, et al.
(författare)
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Integrals, partitions and MacMahon's theorem
- 2007
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Ingår i: Journal of combinatorial theory. Series A (Print). - : Elsevier BV. - 0097-3165 .- 1096-0899. ; 114:3, s. 545-554
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Tidskriftsartikel (refereegranskat)abstract
- In two previous papers, the study of partitions with short sequences has been developed both for its intrinsic interest and for a variety of applications. The object of this paper is to extend that study in various ways. First, the relationship of partitions with no consecutive integers to a theorem of MacMahon and mock theta functions is explored independently. Secondly, we derive in a succinct manner a relevant definite integral related to the asymptotic enumeration of partitions with short sequences. Finally, we provide the generating function for partitions with no sequences of length K and part exceeding N.
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| 5. |
- Balletti, Gabriele
(författare)
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Connectivity through bounds for the Castelnuovo–Mumford regularity
- 2017
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Ingår i: Journal of combinatorial theory. Series A (Print). - : Elsevier BV. - 0097-3165 .- 1096-0899. ; 147, s. 46-54
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Tidskriftsartikel (refereegranskat)abstract
- In this note we generalize and unify two results on connectivity of graphs: one by Balinsky and Barnette, one by Athanasiadis. This is done through a simple proof using commutative algebra tools. In particular we use bounds for the Castelnuovo-Mumford regularity of their Stanley-Reisner rings. As a result, if Delta is a simplicial d-pseudomanifold and s is the largest integer such that A has a missing face of size s, then the 1-skeleton of Delta is inverted right perpendicular(s)/((s+1)d)inverted left perpendicular-connected. We also show that this value is tight.
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| 6. |
- Balletti, Gabriele, et al.
(författare)
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Families of lattice polytopes of mixed degree one
- 2020
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Ingår i: Journal of combinatorial theory. Series A (Print). - : Elsevier BV. - 0097-3165 .- 1096-0899. ; 173
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Tidskriftsartikel (refereegranskat)abstract
- It has been shown by Soprunov that the normalized mixed volume (minus one) of an n-tuple of n-dimensional lattice polytopes is a lower bound for the number of interior lattice points in the Minkowski sum of the polytopes. He defined n-tuples of mixed degree at most one to be exactly those for which this lower bound is attained with equality, and posed the problem of a classification of such tuples. We give a finiteness result regarding this problem in general dimension n >= 4, showing that all but finitely many n-tuples of mixed degree at most one admit a common lattice projection onto the unimodular simplex Delta(n-1). Furthermore, we give a complete solution in dimension n = 3. In the course of this we show that our finiteness result does not extend to dimension n = 3, as we describe infinite families of triples of mixed degree one not admitting a common lattice projection onto the unimodular triangle Delta(2).
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| 7. |
- Björklund, Michael, et al.
(författare)
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The g-theorem matrices are totally nonnegative
- 2009
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Ingår i: Journal of combinatorial theory. Series A (Print). - : Elsevier BV. - 0097-3165 .- 1096-0899. ; 116, s. 730-732
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Tidskriftsartikel (refereegranskat)abstract
- The g-theorem proved by Billera, Lee, and Stanley states that a sequence is the g-vector of a simplicial polytope if and only if it is an M-sequence. For any d-dimensional simplicial polytope the face vector is gM(d) where M-d is a certain matrix whose entries are sums of binomial coefficients. Bjorner found refined lower and upper bound theorems by showing that the (2 x 2)-minors of M-d are nonnegative. He conjectured that all minors of M-d are nonnegative and that is the result of this note.
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| 8. |
- Braun, B., et al.
(författare)
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r-Stable hypersimplices
- 2018
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Ingår i: Journal of combinatorial theory. Series A (Print). - : Academic Press. - 0097-3165 .- 1096-0899. ; 157, s. 349-388
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Tidskriftsartikel (refereegranskat)abstract
- Hypersimplices are well-studied objects in combinatorics, optimization, and representation theory. For each hypersimplex, we define a new family of subpolytopes, called r-stable hypersimplices, and show that a well-known regular unimodular triangulation of the hypersimplex restricts to a triangulation of each r-stable hypersimplex. For the case of the second hypersimplex defined by the two-element subsets of an n-set, we provide a shelling of this triangulation that sequentially shells each r-stable sub-hypersimplex. In this case, we utilize the shelling to compute the Ehrhart h⁎-polynomials of these polytopes, and the hypersimplex, via independence polynomials of graphs. For one such r-stable hypersimplex, this computation yields a connection to CR mappings of Lens spaces via Ehrhart–MacDonald reciprocity.
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| 9. |
- Brubaker, B., et al.
(författare)
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Colored five-vertex models and Demazure atoms
- 2021
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Ingår i: Journal of Combinatorial Theory Series A. - : Elsevier BV. - 0097-3165 .- 1096-0899. ; 178
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Tidskriftsartikel (refereegranskat)abstract
- Type A Demazure atoms are pieces of Schur functions, or sets of tableaux whose weights sum to such functions. Inspired by colored vertex models of Borodin and Wheeler, we will construct solvable lattice models whose partition functions are Demazure atoms; the proof of this makes use of a Yang-Baxter equation for a colored five-vertex model. As a byproduct, we will construct Demazure atoms on Kashiwara's B-infinity crystal and give new algorithms for computing Lascoux-Schutzenberger keys. (C) 2020 Elsevier Inc. All rights reserved.
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| 10. |
- Brändén, Petter, 1976-, et al.
(författare)
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Finite automata and pattern avoidance in words
- 2005
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Ingår i: Journal of Combinatorial Theory - Series A. - : Elsevier BV. - 1096-0899 .- 0097-3165. ; 110:1, s. 127-145
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Tidskriftsartikel (refereegranskat)abstract
- We say that a word w on a totally ordered alphabet avoids the word v if there are no subsequences in w order-equivalent to v. In this paper we suggest a new approach to the enumeration of words on at most k letters avoiding a given pattern. By studying an automaton which for fixed k generates the words avoiding a given pattern we derive several previously known results for these kind of problems, as well as many new. In particular, we give a simple proof of the formula (Electron. J. Combin. 5(1998) #R15) for exact asymptotics for the number of words on k letters of length n that avoids the pattern 12 ⋯ (ℓ + 1). Moreover, we give the first combinatorial proof of the exact formula (Enumeration of words with forbidden patterns, Ph.D. Thesis, University of Pennsylvania, 1998) for the number of words on k letters of length n avoiding a three letter permutation pattern. © 2004 Elsevier Inc. All rights reserved.
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