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  • 1.
    Alexandersson, Per
    et al.
    Stockholm Univ, Dept Math, Stockholm, Sweden..
    Linusson, Svante
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Potka, Samu
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The cyclic sieving phenomenon on circular Dyck paths2019In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 26, no 4, article id P4.16Article in journal (Refereed)
    Abstract [en]

    We give a q-enumeration of circular Dyck paths, which is a superset of the classical Dyck paths enumerated by the Catalan numbers. These objects have recently been studied by Alexandersson and Panova. Furthermore, we show that this q-analogue exhibits the cyclic sieving phenomenon under a natural action of the cyclic group. The enumeration and cyclic sieving is generalized to Mobius paths. We also discuss properties of a generalization of cyclic sieving, which we call subset cyclic sieving, and introduce the notion of Lyndon-like cyclic sieving that concerns special recursive properties of combinatorial objects exhibiting the cyclic sieving phenomenon.

  • 2.
    Alexandersson, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Sawhney, Mehtaab
    A Major-Index Preserving Map on Fillings2017In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 24, no 4, article id P4.3Article in journal (Refereed)
    Abstract [en]

    We generalize a map by S. Mason regarding two combinatorial models for key polynomials, in a way that accounts for the major index. Furthermore we define a similar variant of this map, that regards alternative models for the modified Macdonald polynomials at t = 0, and thus partially answers a question by J. Haglund. These maps together imply a certain uniqueness property regarding inversion- and coinversion-free fillings. These uniqueness properties allow us to generalize the notion of charge to a non-symmetric setting, thus answering a question by A. Lascoux and the analogous question in the symmetric setting proves a conjecture by K. Nelson.

  • 3.
    Amini, Nima
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Equidistributions of mahonian statistics over pattern avoiding permutations2018In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 25, no 1, article id P1.7Article in journal (Refereed)
    Abstract [en]

    A Mahonian d-function is a Mahonian statistic that can be expressed as a linear combination of vincular pattern functions of length at most d. Babson and Ste- ingrímsson classified all Mahonian 3-functions up to trivial bijections and identified many of them with well-known Mahonian statistics in the literature. We prove a host of Mahonian 3-function equidistributions over permutations in Sn avoiding a single classical pattern in S3. Tools used include block decomposition, Dyck paths and generating functions.

  • 4.
    Björner, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Farley, J. D.
    Chain polynomials of distributive lattices are 75% unimodal2005In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 12, no 1Article in journal (Refereed)
    Abstract [en]

    It is shown that the numbers c(i) of chains of length i in the proper part L\{0, 1} of a distributive lattice L of length l + 2 satisfy the inequalities c(0) <...< c([l/2]) and c([3l.4]) >... > c(l). This proves 75% of the inequalities implied by the Neggers unimodality conjecture.

  • 5.
    Björner, Anders.
    et al.
    KTH, Superseded Departments, Mathematics.
    Wachs, M. L.
    Geometrically constructed bases for homology of partition lattices of types A, B and D2004In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 11, no 2Article in journal (Refereed)
    Abstract [en]

    We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A, B and D. This extends and explains the splitting basis for the homology of the partition lattice given in [20], thus answering a question asked by R. Stanley. More explicitly, the following general technique is presented and utilized. Let A be a central and essential hyperplane arrangement in R-d. Let R-1,..., R-k be the bounded regions of a generic hyperplane section of A. We show that there are induced polytopal cycles rho(Ri) in the homology of the proper part LA of the intersection lattice such that {rho(Ri)}(i=1,...,k) is a basis for (H) over tilde (d-2)((L) over bar (A)). This geometric method for constructing combinatorial homology bases is applied to the Coxeter arrangements of types A, B and D, and to some interpolating arrangements.

  • 6. Bränden, Petter
    Sign-graded posets, unimodality of W-polynomials and the Charney-Davis conjecture2004In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 11, no 2Article in journal (Refereed)
    Abstract [en]

    We generalize the notion of graded posets to what we call sign-graded (labeled) posets. We prove that the W-polynomial of a sign-graded poset is symmetric and unimodal. This extends a recent result of Reiner and Welker who proved it for graded posets by associating a simplicial polytopal sphere to each graded poset. By proving that the W-polynomials of sign-graded posets has the right sign at -1, we are able to prove the Charney-Davis Conjecture for these spheres (whenever they are flag).

  • 7.
    Brändén, Petter
    et al.
    Department of Mathematics, Stockholm University.
    Claesson, A.
    Mesh patterns and the expansion of permutation statistics as sums of permutation patterns2011In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, ISSN 1077-8926, Vol. 18, no 2, p. Paper 5-14Article in journal (Refereed)
    Abstract [en]

    Any permutation statistic f : G -> C may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: f = Sigma(tau)lambda(f)(tau)tau . To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern p = (pi, R) is an occurrence of the permutation pattern pi with additional restrictions specified by R on the relative position of the entries of the occurrence. We show that, for any mesh pattern p = (pi, R), wehave lambda(p)(tau) = (-1)(vertical bar tau vertical bar-vertical bar pi vertical bar)p*(tau) where p* = (pi, R(c)) is the mesh pattern with the same underlying permutation as p but with complementary restrictions. We use this result to expand some well known permutation statistics, such as the number of left-to-right maxima, descents, excedances, fixed points, strong fixed points, and the major index. We also show that alternating permutations, Andre permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally, we provide new natural Mahonian statistics.

  • 8. Chow, T Y
    et al.
    Eriksson, Henrik
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Fan, C K
    Chess tableaux2005In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 11, no 2, p. A3-Article in journal (Refereed)
    Abstract [en]

    A chess tableau is a standard Young tableau in which, for all i and j, the parity of the entry in cell ( i; j) equals the parity of i + j + 1. Chess tableaux were first defined by Jonas Sjostrand in his study of the sign-imbalance of certain posets, and were independently rediscovered by the authors less than a year later in the completely different context of composing chess problems with interesting enumerative properties. We prove that the number of 3 x n chess tableaux equals the number of Baxter permutations of n - 1, as a corollary of a more general correspondence between certain three-rowed chess tableaux and certain three-rowed Dulucq-Guibert nonconsecutive tableaux. The correspondence itself is proved by means of an explicit bijection. We also outline how lattice paths, or rat races, can be used to obtain generating functions for chess tableaux. We conclude by explaining the connection to chess problems, and raising some unanswered questions, e. g., there are striking numerical coincidences between chess tableaux and the Charney-Davis statistic; is there a combinatorial explanation?

  • 9.
    Engström, Alexander
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Dochtermann, Anton
    Techn. Universität , Berlin.
    Algebraic properties of edge ideals via combinatorial topology2009In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 16, no 2Article in journal (Refereed)
    Abstract [en]

    We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature regarding linearity, Betti numbers, and (sequentially) Cohen-Macaulay properties of edge ideals associated to chordal, complements of chordal, and Ferrers graphs, as well as trees and forests. Our approach unifies (and in many cases strengthens) these results and also provides combinatorial/enumerative interpretations of certain algebraic properties. We apply our setup to obtain new results regarding algebraic properties of edge ideals in the context of local changes to a graph (adding whiskers and ears) as well as bounded vertex degree. These methods also lead to recursive relations among certain generating functions of Betti numbers which we use to establish new formulas for the projective dimension of edge ideals. We use only well-known tools from combinatorial topology along the lines of independence complexes of graphs, (not necessarily pure) vertex decomposability, shellability, etc.

  • 10. Eriksen, Niklas
    et al.
    Sjöstrand, Jonas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Equidistributed Statistics on Matchings and Permutations2014In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 21, no 4, article id P4.43Article in journal (Refereed)
    Abstract [en]

    We show that the bistatistic of right nestings and right crossings in matchings without left nestings is equidistributed with the number of occurrences of two certain patterns in permutations, and furthermore that this equidistribution holds when refined to positions of these statistics in matchings and permutations. For this distribution we obtain a non-commutative generating function which specializes to Zagier's generating function for the Fishburn numbers after abelianization. As a special case we obtain proofs of two conjectures of Claesson and Linusson. Finally, we conjecture that our results can be generalized to involving left crossings of matchings too.

  • 11.
    Eriksson, Henrik
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Eriksson, Kimmo
    Conjugacy of Coxeter elements2009In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 16, no 2Article in journal (Refereed)
    Abstract [en]

    For a Coxeter group (W, S), a permutation of the set S is called a Coxeter word and the group element represented by the product is called a Coxeter element. Moving the first letter to the end of the word is called a rotation and two Coxeter elements are rotation equivalent if their words can be transformed into each other through a sequence of rotations and legal commutations. We prove that Coxeter elements are conjugate if and only if they are rotation equivalent. This was known for some special cases but not for Coxeter groups in general.

  • 12.
    Eriksson, Henrik
    et al.
    KTH, School of Computer Science and Communication (CSC).
    Eriksson, Kimmo
    Words with intervening neighbours in infinite Coxeter groups are reduced2010In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 17, no 1, p. N9-Article in journal (Refereed)
    Abstract [en]

    Consider a graph with vertex set S. A word in the alphabet S has the intervening neighbours property if any two occurrences of the same letter are separated by all its graph neighbours. For a Coxeter graph, words represent group elements. Speyer recently proved that words with the intervening neighbours property are reduced if the group is infinite and irreducible. We present a new and shorter proof using the root automaton for recognition of reduced words.

  • 13. Hultman, Axel
    Directed subgraph complexes2004In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 11, no 1Article in journal (Refereed)
    Abstract [en]

    Let G be a directed graph, and let Delta(G)(ACY) be the simplicial complex whose simplices are the edge sets of acyclic subgraphs of G. Similarly, we define Delta(G)(NSC) to be the simplicial complex with the edge sets of not strongly connected subgraphs of G as simplices. We show that Delta(G)(ACY) is homotopy equivalent to the (n-1-k)-dimensional sphere if G is a disjoint union of k strongly connected graphs. Otherwise, it is contractible. If G belongs to a certain class of graphs, the homotopy type of Delta(G)(NSC) is shown to be a wedge of (2n-4)-dimensional spheres. The number of spheres can easily be read off the chromatic polynomial of a certain associated undirected graph. We also consider some consequences related to finite topologies and hyperplane arrangements.

  • 14. Jonsson, Jakob
    Hard squares with negative activity and rhombus tilings of the plane2006In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 13, no 1Article in journal (Refereed)
    Abstract [en]

    Let S-m,S-n be the graph on the vertex set Z(m) x Z(n) in which there is an edge between (a, b) and (c, d) if and only if either (a, b) = (c, d +/- 1) or (a, b) (c +/- 1, d) modulo (m, n). We present a formula for the Euler characteristic of the simplicial complex Sigma(m,n) of independent sets in S-m,S-n. In particular, we show that the unreduced Euler characteristic of Sigma(m,n) vanishes whenever m and n are coprime, thereby settling a conjecture in statistical mechanics due to Fendley, Schoutens and van Eerten. For general m and n, we relate the Euler characteristic of Sigma(m,n) to certain periodic rhombus tilings of the plane. Using this correspondence, we settle another conjecture due to Fendley et al., which states that all roots of det(x(I) - T-m) are roots of unity, where T-m is a certain transfer matrix associated to {Sigma(m,n) : n >= 1}. In the language of statistical mechanics, the reduced Euler characteristic of Sigma(m,n) coincides with minus the partition function of the corresponding hard square model with activity -1.

  • 15.
    Jonsson, Jakob
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hard Squares with Negative Activity on Cylinders with Odd Circumference2009In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 16, no 2Article in journal (Refereed)
    Abstract [en]

    Let C-m,C-n be the graph on the vertex set {1, ..., m} x {0, ..., n-1} in which there is an edge between (a, b) and (c, d) if and only if either (a, b) = (c, d +/- 1) or (a, b) = (c +/- 1, d), where the second index is computed modulo n. One may view C-m,C-n as a unit square grid on a cylinder with circumference n units. For odd n, we prove that the Euler characteristic of the simplicial complex Sigma(m,n) of independent sets in C-m,C-n is either 2 or -1, depending on whether or not gcd(m-1, n) is divisble by 3. The proof relies heavily on previous work due to Thapper, who reduced the problem of computing the Euler characteristic of Sigma(m,n) to that of analyzing a certain subfamily of sets with attractive properties. The situation for even n remains unclear. In the language of statistical mechanics, the reduced Euler characteristic of Sigma(m,n) coincides with minus the partition function of the corresponding hard square model with activity -1.

  • 16.
    Jonsson, Jakob
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Hom complexes of set systems2013In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 20, no 1, p. P4-Article in journal (Refereed)
    Abstract [en]

    A set system is a pair S = (V (S), Delta(S)), where Delta(S) is a family of subsets of the set V(S). We refer to the members of Delta(S) as the stable sets of S. A homomorphism between two set systems S and T is a map f : V (S) -> V(T) such that the preimage under f of every stable set of T is a stable set of S. Inspired by a recent generalization due to Engstrom of Lovasz's Hom complex construction, the author associates a cell complex Hom(S, T) to any two finite set systems S and T. The main goal of the paper is to examine basic topological and homological properties of this cell complex for various pairs of set systems.

  • 17.
    Jonsson, Jakob
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Optimal decision trees on simplicial complexes2005In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 12, no 1Article in journal (Refereed)
    Abstract [en]

    We consider topological aspects of decision trees on simplicial complexes, concentrating on how to use decision trees as a tool in topological combinatorics. By Robin Forman's discrete Morse theory, the number of evasive faces of a given dimension i with respect to a decision tree on a simplicial complex is greater than or equal to the ith reduced Betti number (over any field) of the complex. Under certain favorable circumstances, a simplicial complex admits an optimal decision tree such that equality holds for each i; we may hence read off the homology directly from the tree. We provide a recursive definition of the class of semi-nonevasive simplicial complexes with this property. A certain generalization turns out to yield the class of semi-collapsible simplicial complexes that admit an optimal discrete Morse function in the analogous sense. In addition, we develop some elementary theory about semi-nonevasive and semi-collapsible complexes. Finally, we provide explicit optimal decision trees for several well-known simplicial complexes.

  • 18. Nordenstam, Eric
    et al.
    Young, Benjamin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Domino shuffling on Novak half-hexagons and Aztec half-diamonds2011In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 18, no 1, p. P181-Article in journal (Refereed)
    Abstract [en]

    We explore the connections between the well-studied Aztec Diamond graphs and a new family of graphs called the Half-Hexagons, discovered by Jonathan Novak. In particular, both families of graphs have very simple domino shuffling algorithms, which turn out to be intimately related. This connection allows us to prove an "arctic parabola" theorem for the Half-Hexagons as a corollary of the Arctic Circle theorem for the Aztec Diamond.

  • 19.
    Sjöstrand, Jonas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    The cover pebbling theorem2005In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 12, no 1, p. N22-Article in journal (Refereed)
    Abstract [en]

    For any configuration of pebbles on the nodes of a graph, a pebbling move replaces two pebbles on one node by one pebble on an adjacent node. A cover pebbling is a move sequence ending with no empty nodes. The number of pebbles needed for a cover pebbling starting with all pebbles on one node is trivial to compute and it was conjectured that the maximum of these simple cover pebbling numbers is indeed the general cover pebbling number of the graph. That is, for any configuration of this size, there exists a cover pebbling. In this note, we prove a generalization of the conjecture. All previously published results about cover pebbling numbers for special graphs (trees, hypercubes et cetera) are direct consequences of this theorem. We also prove that the cover pebbling number of a product of two graphs equals the product of the cover pebbling numbers of the graphs.

  • 20.
    Visontai, Mirko
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Some remarks on the joint distribution of descents and inverse descents2013In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 20, no 1, p. P52-Article in journal (Refereed)
    Abstract [en]

    We study the joint distribution of descents and inverse descents over the set of permutations of n letters. Gessel conjectured that the two-variable generating function of this distribution can be expanded in a given basis with nonnegative integer coefficients. We investigate the action of the Eulerian operators that give the recurrence for these generating functions. As a result we devise a recurrence for the coefficients in question but are unable to settle the conjecture. We examine generalizations of the conjecture and obtain a type B analog of the recurrence satisfied by the two-variable generating function. We also exhibit some connections to cyclic descents and cyclic inverse descents. Finally, we propose a combinatorial model for the joint distribution of descents and inverse descents in terms of statistics on inversion sequences.

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