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- Backelin, Jörgen, et al.
(author)
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Bergman
- 2003
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In: Computer Algebra Handbook. - 9783540654667 ; , s. 349-352
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Book chapter (other academic/artistic)
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| 3. |
- Backelin, Jörgen, et al.
(author)
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On a class of power ideals
- 2015
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In: Journal of Pure and Applied Algebra. - : Elsevier BV. - 0022-4049 .- 1873-1376. ; 219:8, s. 3158-3180
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Journal article (peer-reviewed)abstract
- In this paper we study the class of power ideals generated by the k(n) forms (x(0) + xi(g1) x(1) + ... + xi(gn) x(n))((k-1)d) where xi is a fixed primitive kth-root of unity and 0 <= g(j) <= k - 1 for all j. For k = 2, by using a Z(k)(n+1)-grading on C[x(0),..., x(n)], we compute the Hilbert series of the associated quotient rings via a simple numerical algorithm. We also conjecture the extension for k > 2. Via Macaulay duality, those power ideals are related to schemes of fat points with support on the k(n) points [1 : xi(g1) : ... : xi(gn)] in P-n. We compute Hilbert series, Betti numbers and Grobner basis for these 0-dimensional schemes. This explicitly determines the Hilbert series of the power ideal for all k: that this agrees with our conjecture for k > 2 is supported by several computer experiments.
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| 4. |
- Backelin, Jörgen, et al.
(author)
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Parity splits by triple point distances in X-trees
- 2006
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In: Annals of Combinatorics. - : Springer Science and Business Media LLC. - 0218-0006 .- 0219-3094. ; 10:1, s. 1-18
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Journal article (peer-reviewed)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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| 8. |
- Backelin, Jörgen, et al.
(author)
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Wilf-equivalence for singleton classes
- 2007
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In: Advances in Applied Mathematics. - : Elsevier BV. - 0196-8858 .- 1090-2074. ; 38:2, s. 133-148
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Journal article (peer-reviewed)abstract
- Write p(1)p(2)(...)pm for the permutation matrix (delta p(i) j)(m x m). Let S-n(M) be the set of n x n permutation matrices which do not contain the m x m permutation matrix M as a submatrix. In [R. Simion, F.W. Schmidt, Restricted permutations, European J. Combin. 6 (1985) 383-406] Simion and Schmidt show bijectively that vertical bar S-n (123)vertical bar = vertical bar S-n (213)vertical bar. In the present work, we give a bi jection from S-n (12...tp(t+1)... p(m)) to S-n (t...21 p(t+1)...p(m)). This result was established for t = 2 in [J. West, Permutations with forbidden subsequences and stack-sortable permutations, PhD thesis, MIT, Cambridge, MA, 1990] and for t = 3 in [E. Babson, J. West, The permutations 123p(4)... p(t) and 321 p(4)...p(t) are Wilf-equivalent, Graphs Combin. 16 (2001) 373-3801. Moreover, if we think of n x n permutation matrices as transversals of the n by n square diagram, then we generalise this result to transversals of Young diagrams.
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| 9. |
- Beyene, Fufa, et al.
(author)
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Set Partitions and Other Bell Number Enumerated Objects
- 2023
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In: Journal of Integer Sequences. - 1530-7638. ; 26:1
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Journal article (peer-reviewed)abstract
- In this paper, we study classes of subexcedant functions enumerated by the Bell numbers and present bijections on set partitions. We present a set of permutations whose transposition arrays are the restricted growth functions, thus defining Bell permutations of the second kind. We describe a bijection between Bell permutations of the first kind (introduced by Ponti and Vajnovzski) and the second kind. We present two other Bell number enumerated classes of subexcedant functions. Further, we present bijections on set partitions, in particular, an involution that interchanges the set of merging blocks and the set of successions. We use the bijections to enumerate the distribution of these statistics over the set of set partitions, and also give some enumeration results.
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| 10. |
- Emtander, Eric, 1979-
(author)
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Chordal and Complete Structures in Combinatorics and Commutative Algebra
- 2010
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Doctoral thesis (other academic/artistic)abstract
- This thesis is divided into two parts. The first part is concerned with the commutative algebra of certain combinatorial structures arising from uniform hypergraphs. The main focus lies on two particular classes of hypergraphs called chordal hypergraphs and complete hypergraphs, respectively. Both these classes arise naturally as generalizations of the corresponding well known classes of simple graphs. The classes of chordal and complete hypergraphs are introduced and studied in Chapter 2 and Chapter 3 respectively. Chapter 4, that is the content of \cite{E5}, answers a question posed at the P.R.A.G.MAT.I.C. summer school held in Catania, Italy, in 2008. In Chapter 5 we study hypergraph analogues of line graphs and cycle graphs. Chapter 6 is concerned with a connectedness notion for hypergraphs and in Chapter 7 we study a weak version of shellability.The second part is concerned with affine monoids and their monoid rings. Chapter 8 provide a combinatorial study of a class of positive affine monoids that behaves in some sense like numerical monoids. Chapter 9 is devoted to the class of numerical monoids of maximal embedding dimension. A combinatorial description of the graded Betti numbers of the corresponding monoid rings in terms of the minimal generators of the monoids is provided. Chapter 10 is concerned with monomial subrings generated by edge sets of complete hypergraphs.
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