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  • 1. Andrews, George
    et al.
    Eriksson, Henrik
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Petrov, Fedor
    Romik, Dan
    Integrals, partitions and MacMahon's theorem2007In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 114, no 3, p. 545-554Article in journal (Refereed)
    Abstract [en]

    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.

  • 2.
    Björklund, Michael
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Engström, Alexander
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    The g-theorem matrices are totally nonnegative2009In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 116, p. 730-732Article in journal (Refereed)
    Abstract [en]

    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.

  • 3.
    Björner, Anders
    KTH, Superseded Departments, Mathematics.
    Nerves, fibers and homotopy groups2003In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 102, no 1, p. 88-93Article in journal (Refereed)
    Abstract [en]

    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.

  • 4. Braun, B.
    et al.
    Solus, Liam
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    r-Stable hypersimplices2018In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 157, p. 349-388Article in journal (Refereed)
    Abstract [en]

    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.

  • 5. Brändén, Petter
    et al.
    Mansour, Toufik
    Finite automata and pattern avoidance in words2005In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 110, no 1, p. 127-145Article in journal (Refereed)
    Abstract [en]

    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.

  • 6.
    Goodarzi, Afshin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Clique vectors of k-connected chordal graphs2015In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 132, p. 188-193Article in journal (Refereed)
    Abstract [en]

    The clique vector c(G) of a graph G is the sequence (c(1), c(2), ..., c(d)) in N-d, where c(i) is the number of cliques in G with i vertices and d is the largest cardinality of a clique in G. In this note, we use tools from commutative algebra to characterize all possible clique vectors of k-connected chordal graphs.

  • 7.
    Goodarzi, Afshin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the Hilbert series of monomial ideals2013In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 120, no 2, p. 315-317Article in journal (Refereed)
    Abstract [en]

    To every squarefree monomial ideal one can associate a hypergraph. In this paper we show that the Hilbert series of a squarefree monomial ideal can be obtained from the so-called edge induced polynomial of the associated hypergraph.

  • 8.
    Hultman, Axel
    KTH, Superseded Departments, Mathematics.
    Bruhat intervals of length 4 in Weyl groups2003In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 102, no 1, p. 163-178Article in journal (Refereed)
    Abstract [en]

    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.

  • 9.
    Hultman, Axel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Linusson, Svante
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shareshian, John
    Sjostrand, Jonas
    From Bruhat intervals to intersection lattices and a conjecture of Postnikov2009In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 116, no 3, p. 564-580Article in journal (Refereed)
    Abstract [en]

    We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation w is an element of (sic)(n). is at most the number of elements below w in the Bruhat order, and (B) that equality holds if and only if w avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups. A byproduct of this result and its proof is a set of inequalities relating Betti numbers of complexified inversion arrangements to Betti numbers of closed Schubert cells. Another consequence is a simple combinatorial interpretation of the chromatic polynomial of the inversion graph of a permutation which avoids the above patterns.

  • 10.
    Incitti, Federico
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    More on the combinatorial invariance of Kazhdan-Lusztig polynomials2007In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 114, no 3, p. 461-482Article in journal (Refereed)
    Abstract [en]

    We prove that the Kazhdan-Lusztig polynomials are combinatorial invariants for intervals up to length 8 in Coxeter groups of type A and up to length 6 in Coxeter groups of type B and D. As a consequence of our methods, we also obtain a complete classification, up to isomorphism, of Bruhat intervals of length 7 in type A and of length 5 in types B and D, which are not lattices.

  • 11.
    Jonsson, Jakob
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Exact sequences for the homology of the matching complex2008In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 115, no 8, p. 1504-1526Article in journal (Refereed)
    Abstract [en]

    Building on work by Bouc and by Shareshian and Wachs, we provide a toolbox of long exact sequences for the reduced simplicial homology of the matching complex M., which is the simplicial complex of matchings in the complete graph K-n. Combining these sequences in different ways, we prove several results about the 3-torsion part of the homology of M, First, we demonstrate that there is nonvanishing 3-torsion in (H) over bar (d)(M-n : Z) whenever v(n) <= d <= [n-6/2], where v(n) =[n-4/3]. By results due to Bouc and to Shareshian and Wachs, (H) over bar (d)(M-n : Z) is a nontrivial elementary 3-group for almost all n and the bottom nonvanishing homology group of M. for all n 0 2. Second, we prove that (H) over bar (d)(M-n : Z) is a nontrivial 3-group whenever v(n) <= d <= [2n-9/5]. Third, for each k >= 0, we show that there is a polynomial f(k)(r) of degree 3k such that the dimension of (H) over bar (k-1+r) (M2k+1+3r:Z(3)), viewed as a vector space over Z(3), is at most f(k)(r) for all r >= k + 2.

  • 12.
    Jonsson, Jakob
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Generalized triangulations and diagonal-free subsets of stack polyrominoes2005In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 112, no 1, p. 117-142Article in journal (Refereed)
    Abstract [en]

    For n >= 3, let Omega(n) be the set of line segments between vertices in a convex n-gon. For j >= 1, a j-crossing is a set of j distinct and mutually intersecting line segments from Q,, such that all 2j endpoints are distinct. For k >= 1, let Delta(n,k) be the simplicial complex of subsets of Omega(n) not containing any (k + 1)-crossing. For example, Delta(n.1) has one maximal set for each triangulation of the n-gon. Dress, Koolen, and Moulton were able to prove that all maximal sets in Delta(n,k) have the same number k(2n - 2k - 1) of line segments. We demonstrate that the number of such maximal sets is counted by a k x k determinant of Catalan numbers. By the work of Desainte-Catherine and Viennot, this determinant is known to count quite a few other objects including fans of non-crossing Dyck paths. We generalize our result to a larger class of simplicial complexes including some of the complexes appearing in the work of Herzog and Trung, on determinantal ideals.

  • 13. Jonsson, Jakob
    On the topology of simplicial complexes related to 3-connected and Hamiltonian graphs2003In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 104, no 1, p. 169-199Article in journal (Refereed)
    Abstract [en]

    Using techniques from Robin Forman's discrete Morse theory, we obtain information about the homology and homotopy type of some graph complexes. Specifically, we prove that the simplicial complex Delta(n)(3) of not 3-connected graphs on it vertices is homotopy equivalent to a wedge of (n - 3) (.) (n - 2)!/2 spheres of dimension 2n - 4, thereby verifying a conjecture by Babson, Bjorner, Linusson, Shareshian, and Welker. We also determine a basis for the corresponding nonzero homology group in the CW complex of 3-connected graphs. In addition, we show that the complex Gamma(n) of non-Hamiltonian graphs on it vertices is homotopy equivalent to a wedge of two complexes, one of the complexes being the complex Delta(n)(2) of not 2-connected graphs on it vertices. The homotopy type of Delta(n)(2) has been determined, independently, by the five authors listed above and by Turchin. While Gamma(n) and Delta(n)(2) are homotopy equivalent for small values on it, they are nonequivalent for n = 10.

  • 14.
    Sjöstrand, Jonas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Bruhat intervals as rooks on skew Ferrers boards2007In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 114, no 7, p. 1182-1198Article in journal (Refereed)
    Abstract [en]

    We characterise the permutations pi such that the elements in the closed lower Bruhat interval [id, pi] of the symmetric group correspond to non-taking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations pi such that [id, pi] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner.Our characterisation connect, the Poincare polynomials (rank-generating function) of Bruhat intervals with q-rook polynomials, and we are able to compute the Poincare polynomial of some particularly interesting intervals in the finite Weyl groups An and B, The expressions involve q-Stirling numbers of the second kind, and for the group A, putting q = 1 yields the poly-Bernoulli numbers defined by Kaneko.

  • 15.
    Sjöstrand, Jonas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    On the sign-imbalance of partition shapes2005In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 111, no 2, p. 190-203Article in journal (Refereed)
    Abstract [en]

    Let the sign of a standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. A conjecture by Richard Stanley says that the sum of the signs of all SYTs with n squares is 2([n/2]). We present a stronger theorem with a purely combinatorial proof using the Robinson-Schensted correspondence and a new concept called chess tableaux. We also prove a sharpening of another conjecture by Stanley concerning weighted sums of squares of sign-imbalances. The proof is built on a remarkably simple relation between the sign of a permutation and the signs of its RS-corresponding tableaux.

  • 16.
    Visontai, Mirko
    et al.
    University of Pennsylvania, United States.
    Williams, Nathan
    Stable multivariate W-Eulerian polynomials2013In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 120, no 7, p. 1929-1945Article in journal (Refereed)
    Abstract [en]

    We prove a multivariate strengthening of Brenti's result that every root of the Eulerian polynomial of type B is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability a generalization of real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator. Our results extend naturally to colored permutations, and we also give stable generalizations of recent real-rootedness results due to Dilks, Petersen, and Stembridge on affine Eulerian polynomials of types A and C. Finally, although we are not able to settle Brenti's real-rootedness conjecture for Eulerian polynomials of type D. nor prove a companion conjecture of Dilks, Petersen, and Stembridge for affine Eulerian polynomials of types B and D. we indicate some methods of attack and pose some related open problems.

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