| 1. |
- Alexandersson, Per, et al.
(författare)
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Some conjectures on the Schur expansion of Jack polynomials
- 2021
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Ingår i: Journal of Combinatorics. - : INT PRESS BOSTON, INC. - 2156-3527 .- 2150-959X. ; 12:2, s. 215-233
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Tidskriftsartikel (refereegranskat)abstract
- We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found in these bases, including Eulerian numbers, Stirling numbers, quasi-Yamanouchi tableaux, and rook boards. These results also lead to further conjectures about the fundamental quasisymmetric expansions of these bases, which we prove for special cases.
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| 2. |
- Bränden, Petter, 1976-, et al.
(författare)
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Lecture hall P-partitions
- 2020
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Ingår i: Journal of Combinatorics. - : INT PRESS BOSTON, INC. - 2156-3527 .- 2150-959X. ; 11:2, s. 391-412
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Tidskriftsartikel (refereegranskat)abstract
- We introduce and study s-lecture hall P-partitions which is a generalization of s-lecture hall partitions to labeled (weighted) posets. We provide generating function identities for s-lecture hall P-partitions that generalize identities obtained by Savage and Schuster for s-lecture hall partitions, and by Stanley for P-partitions. We also prove that the corresponding (P, s)-Eulerian polynomials are real-rooted for certain pairs (P, s), and speculate on unimodality properties of these polynomials.
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| 3. |
- Casselgren, Carl Johan, 1982-, et al.
(författare)
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Completing partial Latin squares with two filled rows and three filled columns
- 2023
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Ingår i: Journal of Combinatorics. - : INT PRESS BOSTON, INC. - 2156-3527 .- 2150-959X. ; 14:1, s. 139-153
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Tidskriftsartikel (refereegranskat)abstract
- Consider a partial Latin square P where the first two rows and first three columns are completely filled, and every other cell of P is empty. It has been conjectured that all such partial Latin squares of order at least 8 are completable. Based on a technique by Kuhl and McGinn we describe a framework for completing partial Latin squares in this class. Moreover, we use our method for proving that all partial Latin squares from this family, where the intersection of the nonempty rows and columns form a Latin rectangle with three distinct symbols, are completable.
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| 4. |
- Falgas-Ravry, Victor, et al.
(författare)
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Separating path systems
- 2014
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Ingår i: Journal of Combinatorics. - 2156-3527 .- 2150-959X. ; 5:3, s. 335-354
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Tidskriftsartikel (refereegranskat)abstract
- We study separating systems of the edges of a graph where each member of the separating system is a path. We conjecture that every nn-vertex graph admits a separating path system of size linear in nn and we prove this in certain interesting special cases. In particular, we establish this conjecture for random graphs and graphs with linear minimum degree. We also obtain tight bounds on the size of a minimal separating path system in the case of trees.
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| 5. |
- Hägglund, Jonas, et al.
(författare)
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Shortest cycle covers and cycle double covers with large 2-regular subgraphs
- 2013
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Ingår i: Journal of Combinatorics. - 2156-3527 .- 2150-959X. ; 4:4, s. 457-468
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we show that many snarks have a shortest cycle cover of length 4/3 m + c for a constant c, where m is the number of edges in the graph, in agreement with the conjecture that all snarks have shortest cycle covers of length 4/3 m + o(m). In particular, we prove that graphs with perfect matching index at most 4 have cycle covers of length 4/3 m and satisfy the (1,2)-covering conjecture of Zhang, and that graphs with large circumference have cycle covers of length close to 4/3 m. We also prove some results for graphs with low oddness and discuss the connection with Jaeger’s Petersen colouring conjecture.
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| 6. |
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| 7. |
- Janson, Svante, 1955-, et al.
(författare)
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On the asymptotic statistics of the number of occurrences of multiple permutation patterns
- 2015
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Ingår i: Journal of Combinatorics. - 2156-3527 .- 2150-959X. ; 6:1-2, s. 117-143
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Tidskriftsartikel (refereegranskat)abstract
- We study statistical properties of the random variables Xσ(π), the number of occurrences of the pattern σ in the permutation π. We present two contrasting approaches to this problem: traditional probability theory and the “less traditional” computational approach. Through the perspective of the first approach, we prove that for any pair of patterns σ and τ, the random variables Xσ and Xτ are jointly asymptotically normal (when the permutation is chosen from Sn). From the other perspective, we develop algorithms that can show asymptotic normality and joint asymptotic normality (up to a point) and derive explicit formulas for quite a few moments and mixed moments empirically, yet rigorously. The computational approach can also be extended to the case where permutations are drawn from a set of pattern avoiders to produce many empirical moments and mixed moments. This data suggests that some random variables are not asymptotically normal in this setting.
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| 8. |
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| 9. |
- Linusson, Svante, et al.
(författare)
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Properties of the Edelman-Greene bijection
- 2020
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Ingår i: Journal of Combinatorics. - Somerville, MA. - 2156-3527 .- 2150-959X. ; 11:2, s. 249-273
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Tidskriftsartikel (refereegranskat)abstract
- Edelman and Greene constructed a bijective correspondence between reduced words of the reverse permutation and standard Young tableaux. We prove that for any reduced word the shape of the region of the insertion tableau containing the smallest possible entries evolves exactly as the upper-left component of the permutation's (Rothe) diagram. Properties of the Edelman-Greene bijection restricted to 132-avoiding and 2143-avoiding permutations are presented. We also consider the Edelman-Greene bijection applied to non-reduced words.
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| 10. |
- Linusson, Svante, et al.
(författare)
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Rees products and lexicographic shellability
- 2012
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Ingår i: Journal of Combinatorics. - 2156-3527 .- 2150-959X. ; 3:3, s. 243-276
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Tidskriftsartikel (refereegranskat)abstract
- We use the theory of lexicographic shellability to provide various examples in which the rank of the homology of a Rees product of two partially ordered sets enumerates some set of combinatorial objects, perhaps according to some natural statistic on the set. Many of these examples generalize a result of J. Jonsson, which says that the rank of the unique nontrivial homology group of the Rees product of a truncated Boolean algebra of degree n and a chain of length n−1 is the number of derangements in Sn.
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