Showing papers in "Annals of Combinatorics in 2010"
TL;DR: It is shown that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoped solution, proves algebraic independence of certain types of sums, and summation-theory shows transcendence of whole classes of sums.
Abstract: Usually creative telescoping is used to derive recurrences for sums. In this article we show that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoping solution, proves algebraic independence of certain types of sums. Combining this fact with summation-theory shows transcendence of whole classes of sums. Moreover, this result throws new light on the question why, for example, Zeilberger’s algorithm fails to find a recurrence with minimal order.
89 citations
TL;DR: In this article, a sufficient condition for a graph being Hamiltonian is that the nontrivial eigenvalues of the combinatorial Laplacian are sufficiently close to the average degree of the graph.
Abstract: We consider the spectral and algorithmic aspects of the problem of finding a Hamil- tonian cycle in a graph. We show that a sufficient condition for a graph being Hamiltonian is that the nontrivial eigenvalues of the combinatorial Laplacian are sufficiently close to the average degree of the graph. An algorithm is given for the problem of finding a Hamiltonian cycle in graphs with bounded spectral gaps which has complexity of order n c ln n .
47 citations
TL;DR: In this paper, a nice combinatorial formula for characters of irreducible representations of the symmetric group of rectangular shape is given. But it is not a generalization for any shape.
Abstract: In his paper [9], Stanley finds a nice combinatorial formula for characters of irreducible representations of the symmetric group of rectangular shape. Then, in [10], he gives a conjectural generalisation for any shape. Here, we will prove this formula using shifted Schur functions and Jucys-Murphy elements.
36 citations
TL;DR: In this article, the first one of a series of papers on association of orientations, lattice polytopes, and group arrangements to graphs is presented, where the purpose is to interpret the integral and modular tension polynomials of graphs at zero and negative integers.
Abstract: This is the first one of a series of papers on association of orientations, lattice polytopes, and group arrangements to graphs. The purpose is to interpret the integral and modular tension polynomials of graphs at zero and negative integers. The whole exposition is put under the framework of subgroup arrangements and the application of Ehrhart polynomials. Such a viewpoint leads to the following main results of the paper: (i) the reciprocity law for integral tension polynomials; (ii) the reciprocity law for modular tension polynomials; and (iii) a new interpretation for the value of the Tutte polynomial T(G; x, y) of a graph G at (1, 0) as the number of cut-equivalence classes of acyclic orientations on G.
33 citations
TL;DR: In this article, a partial order on the set of skew characters of the symmetric group is introduced, which is used to classify the multiplicity-free skew characters and give a short and easy proof that the Schubert calculus is equivalent to the skew characters.
Abstract: In this paper we introduce a partial order on the set of skew characters of the symmetric group which we use to classify the multiplicity-free skew characters. Furthermore, we give a short and easy proof that the Schubert calculus is equivalent to that of skew characters in the following sense: If we decompose the product of two Schubert classes we get the same as if we decompose a skew character and replace the irreducible characters by Schubert classes of the ‘inverse’ partitions (Theorem 4.3).
28 citations
TL;DR: Lower and upper bounds for the number of permutations of length n generated by two stacks in series, two stack in parallel, and a general deque were given in this paper.
Abstract: Lower and upper bounds are given for the the number of permutations of length n generated by two stacks in series, two stacks in parallel, and a general deque.
26 citations
TL;DR: The Tchebyshev transform of the first and second kind of a poset is studied in this article, which preserves desirable combinatorial properties, including EL-shellability and nonnegativity of the cd-index.
Abstract: An in-depth study of the Tchebyshev transforms of the first and second kind of a poset is taken. The Tchebyshev transform of the first kind is shown to preserve desirable combinatorial properties, including EL-shellability and nonnegativity of the cd-index. When restricted to Eulerian posets, it corresponds to the Billera, Ehrenborg, and Readdy omega map of oriented matroids. The Tchebyshev transform of the second kind U is a Hopf algebra endomorphism on the space of quasisymmetric functions which, when restricted to Eulerian posets, coincides with Stembridge’s peak enumerator. The complete spectrum of U is determined, generalizing the work of Billera, Hsiao, and van Willigenburg. The type B quasisymmetric function of a poset is introduced and, like Ehrenborg’s classical quasisymmetric function of a poset, it is a comodule morphism with respect to the quasisymmetric functions QSym. Finally, similarities among the omega map, Ehrenborg’s r-signed Birkhoff transform, and the Tchebyshev transforms motivate a general study of chain maps which occur naturally in the setting of combinatorial Hopf algebras.
24 citations
TL;DR: In this paper, the colored quasisymmetric functions are shown to form a Hopf algebra with a natural peak subalgebra, which is the image of the algebra of colored posets.
Abstract: The colored quasisymmetric functions, like the classic quasisymmetric functions, are known to form a Hopf algebra with a natural peak subalgebra. We show how these algebras arise as the image of the algebra of colored posets. To effect this approach, we introduce colored analogs of P-partitions and enriched P-partitions. We also frame our results in terms of Aguiar, Bergeron, and Sottile's theory of combinatorial Hopf algebras and its colored analog.
23 citations
TL;DR: In this article, the asymptotic probability that two randomly selected compositions of n into parts equal to a or b have the same number of parts has been derived for parts of sizes 1 and 2 with weighted lattice paths and central Whitney numbers of fence posets.
Abstract: We compute the asymptotic probability that two randomly selected compositions of n into parts equal to a or b have the same number of parts In addition, we provide bijections in the case of parts of sizes 1 and 2 with weighted lattice paths and central Whitney numbers of fence posets Explicit algebraic generating functions and asymptotic probabilities are also computed in the case of pairs of compositions of n into parts at least d, for any fixed natural number d
23 citations
TL;DR: In this paper, the problem of enumerating periodic σ-juggling sequences of length n for multiplex juggling, where σ is the initial state (or landing schedule) of the balls, was studied.
Abstract: We consider the problem of enumerating periodic σ-juggling sequences of length n for multiplex juggling, where σ is the initial state (or landing schedule) of the balls. We first show that this problem is equivalent to choosing 1’s in a specified matrix to guarantee certain column and row sums, and then using this matrix, derive a recursion. This work is a generalization of earlier work of Chung and Graham.
20 citations
TL;DR: In this article, the authors state and prove q-analogues of several Pell identities via weighted tilings, and show that these identities can be interpreted as enumerators of certain sets of tilings using white squares, black squares, and gray dominoes.
Abstract: Recently, Benjamin, Plott, and Sellers proved a variety of identities involving sums of Pell numbers combinatorially by interpreting both sides of a given identity as enumerators of certain sets of tilings using white squares, black squares, and gray dominoes. In this article, we state and prove q-analogues of several Pell identities via weighted tilings.
TL;DR: In this paper, the authors characterise and enumerate permutations that are sortable by n − 4 passes through a stack and conjecture the number of permutations sortable in n − 5 passes, and also the form of a formula for the general case n − k.
Abstract: We characterise and enumerate permutations that are sortable by n – 4 passes through a stack. We conjecture the number of permutations sortable by n – 5 passes, and also the form of a formula for the general case n – k, which involves a polynomial expression.
TL;DR: In this paper, the authors obtained a generating function for the number of permutations with no τ-match starting at position i, if the elements in positions i, i+1,..., i+j−1 in σ have the same relative order as the elements of τ.
Abstract: Given a permutation τ of length j, we say that a permutation σ has a τ-match starting at position i, if the elements in positions i, i+1, . . . , i+j−1 in σ have the same relative order as the elements of τ. We have been able to take advantage of the results of Mendes and Remmel [1] to obtain a generating function for the number of permutations with no τ-matches for several new classes of permutations. These new classes include a large class of permutations which are shuffles of an increasing sequence 1 2 · · · n with an arbitrary permutation σ of the integers {n + 1, . . . , n + m}. Finally we give a formula for the generating function for the number of permutations which have no 1 3 2 4-matches.
TL;DR: In this paper, the (isotropic) unitary graph (U \left(n, q 2, q 3 + 1) is introduced, and the group of graph automorphisms of unitary graphs with n ≥ 4, 5 is determined.
Abstract: The (isotropic) unitary graph \({U \left(n, q^{2}\right)}\) is introduced. When n = 2 or 3, \({U \left(2, q^{2}\right)}\) or \({U \left(3, q^{2}\right)}\) are complete graphs with q + 1 or q3 + 1 vertices, respectively. When n ≥ 4, it is shown that \({U \left(n, q^{2}\right)}\) is strongly regular and its parameters are computed. The group of graph automorphisms of \({U \left(n, q^{2}\right)}\) , when n ≠ 4, 5, is determined.
TL;DR: In this article, the Robinson-Schensted-Knuth correspondence and Schutzenberger's evacuation of standard tableaux were used to enumerate permutations and involutions which are invariant under the reverse-complement map and which have no decreasing subsequences of length k.
Abstract: We use the Robinson-Schensted-Knuth correspondence and Schutzenberger’s evacuation of standard tableaux to enumerate permutations and involutions which are invariant under the reverse-complement map and which have no decreasing subsequences of length k. These enumerations are in terms of numbers of permutations with no decreasing subsequences of length approximately $${{\frac{k}{2}};}$$
we use known results concerning these quantities to give explicit formulas when k ≤ 6.
TL;DR: In this paper, the authors characterize separable multidimensional permutations in terms of forbidden patterns and enumerate them by means of generating function, recursive formula, and explicit formula.
Abstract: We characterize separable multidimensional permutations in terms of forbidden patterns and enumerate them by means of generating function, recursive formula, and explicit formula. We find a connection between multidimensional permutations and guillotine partitions of a box. In particular, a bijection between separable d-dimensional permutations and guillotine partitions of a 2d-1-dimensional box is constructed. We also study enumerating problems related to guillotine partitions under certain restrictions revealing connections to other combinatorial structures. This allows us to obtain several results on patterns in permutations.
TL;DR: In this article, it was shown that for any fixed r and sufficiently large n, there is a monochromatic Berge-cycle of length (1 − o(1))n in every (r − 1)-coloring of the edges of the complete r-uniform hypergraph on n vertices.
Abstract: Edge colorings of r-uniform hypergraphs naturally define a multicoloring on the 2-shadow, i.e., on the pairs that are covered by hyperedges. We show that in any (r – 1)-coloring of the edges of an r-uniform hypergraph with n vertices and at least \((1-\varepsilon)\left( {\begin{array}{*{20}c} n\\ r\\ \end{array}}\right)\) edges, the 2-shadow has a monochromatic matching covering all but at most o(n) vertices. This result confirms an earlier conjecture and implies that for any fixed r and sufficiently large n, there is a monochromatic Berge-cycle of length (1 – o(1))n in every (r – 1)-coloring of the edges of \({K^{(r)}_{n}}\), the complete r-uniform hypergraph on n vertices.
TL;DR: In this article, a map relating hyper-octahedral Robinson-Schensted algorithms on standard domino tableaux of unequal rank is described, and iterated of this map relates the algorithms defined by Garfinkle and Stanton-White and when restricted to involutions.
Abstract: We describe a map relating hyperoctahedral Robinson-Schensted algorithms on standard domino tableaux of unequal rank. Iteration of this map relates the algorithms defined by Garfinkle and Stanton-White and when restricted to involutions, this construction answers a question posed by van Leeuwen. The principal technique is derived from operations defined on standard domino tableaux by Garfinkle which must be extended to this more general setting.
TL;DR: In this paper, the multiplicity-free representations of the symmetric group were determined by a combinatorial problem involving systems of set-partitions with a specific pattern of intersection.
Abstract: We determine all the multiplicity-free representations of the symmetric group. This project is motivated by a combinatorial problem involving systems of set-partitions with a specific pattern of intersection.
TL;DR: In this paper, the parity alternating permutations with even and odd entries alternately were introduced and their combinatorial properties were investigated. But their properties were not discussed in detail, except for the relation between parity alternating and signed Eulerian numbers.
Abstract: This paper introduces subgroups of the symmetric group and studies their combinatorial properties. Their elements are called parity alternating, because they are permutations with even and odd entries alternately. The objective of this paper is twofold. The first is to derive several properties of such permutations by subdividing them into even and odd permutations. The second is to discuss their combinatorial properties; among others, relationships between those permutations and signed Eulerian numbers. Divisibility properties by prime powers are also deduced for signed Eulerian numbers and several related numbers.
TL;DR: In this article, random walk on the irreducible representations of the symmetric and general linear groups is studied and a separation distance cutoff is proved and the exact separation distance asymptotics are determined.
Abstract: Random walk on the irreducible representations of the symmetric and general linear groups is studied. A separation distance cutoff is proved and the exact separation distance asymptotics are determined. A key tool is a method for writing the multiplicities in the Kronecker tensor powers of a fixed representation as a sum of non-negative terms. Connections are made with the Lagrange-Sylvester interpolation approach to Markov chains.
TL;DR: For the chessboard complex Mm,n, a simplicial complex of matchings in the complete bipartite graph Km, n, the authors showed that there is a polynomial fk(a, b) of degree 3k such that the dimension of Mm-n is at most O(k+a+2b-2) for all a ≥ 0 and b ≥ k + 2.
Abstract: Let 1 ≤ m ≤ n. We prove various results about the chessboard complex Mm,n, which is the simplicial complex of matchings in the complete bipartite graph Km,n. First, we demonstrate that there is nonvanishing 3-torsion in \({{\tilde{H}_d({\sf M}_{m,n}; {\mathbb Z})}}\) whenever \({{\frac{m+n-4}{3}\leq d \leq m-4}}\) and whenever 6 ≤ m < n and d = m − 3. Combining this result with theorems due to Friedman and Hanlon and to Shareshian and Wachs, we characterize all triples (m, n, d ) satisfying \({{\tilde{H}_d \left({\sf M}_{m,n}; {\mathbb Z}\right)
eq 0}}\). Second, for each k ≥ 0, we show that there is a polynomial fk(a, b) of degree 3k such that the dimension of \({{\tilde{H}_{k+a+2b-2}}\,\left({{\sf M}_{k+a+3b-1,k+2a+3b-1}}; \mathbb Z_{3}\right)}\), viewed as a vector space over \({\mathbb{Z}_3}\), is at most fk(a, b) for all a ≥ 0 and b ≥ k + 2. Third, we give a computer-free proof that \({{\tilde{H}_2 ({\sf M}_{5,5}; \mathbb {Z})\cong \mathbb Z_{3}}}\). Several proofs are based on a new long exact sequence relating the homology of a certain subcomplex of Mm,n to the homology of Mm-2,n-1 and Mm-2,n-3.
TL;DR: In this article, it was shown that the number of nonzero digits of u====== n====== in base g is large on a set of n of asymptotic density 1.
Abstract: Let g ≥ 2 be an integer and let (u
n
)n≥1 be a sequence of integers which satisfies a relation u
n+1 = h(n)u
n
for a rational function h(X). For example, various combinatorial numbers as well as their products satisfy relations of this type. Here, we show that under some mild technical assumptions the number of nonzero digits of u
n
in base g is large on a set of n of asymptotic density 1. We also extend this result to a class of sequences satisfying relations of second order u
n+2 = h
1(n)u
n+1 + h
2(n)u
n
with two nonconstant rational functions $${h_1(X), h_2(X) \in {\mathbb Q} [X]}$$
. This class includes the Apery, Delannoy, Motzkin, and Schroder numbers.
TL;DR: In this article, a generalization of Dyson's notion of rank to integer partitions with k successive Durfee rectangles is presented and two combinatorial symmetries associated with this new definition are given.
Abstract: We present a generalization, which we call (k, m)-rank, of Dyson’s notion of rank to integer partitions with k successive Durfee rectangles and give two combinatorial symmetries associated with this new definition. We prove these symmetries bijectively. Using the two symmetries we give a new combinatorial proof of generalized Rogers-Ramanujan identities. We also describe the relationship between (k, m)-rank and Garvan’s k-rank.
TL;DR: This paper provides a recursive method to calculate the size of the unit-neighbourhood for any tree in the space of unrooted binary phylogenetic trees with n-leaves, and gives both upper and lower bounds on this size.
Abstract: In this paper, we study the unit-neighbourhood of the tree bisection and reconnection operation on unrooted binary phylogenetic trees. Specifically, we provide a recursive method to calculate the size of the unit-neighbourhood for any tree in the space \({\fancyscript{T}_n}\) of unrooted binary phylogenetic trees with n-leaves. We also give both upper and lower bounds on this size for all trees in \({\fancyscript{T}_n}\), and characterize those trees for which the stated upper bound is sharp.
TL;DR: In this article, the authors use combinatorial techniques to enumerate symmetric permutations which avoid one pattern of length three and one patterns of length four. But they do not enumerate permutations that are invariant under actions of certain subgroups of the symmetry group of a square.
Abstract: Symmetric pattern-avoiding permutations are restricted permutations which are invariant under actions of certain subgroups of D4, the symmetry group of a square. We examine pattern-avoiding permutations with 180° rotational-symmetry. In particular, we use combinatorial techniques to enumerate symmetric permutations which avoid one pattern of length three and one pattern of length four. Our results involve well-known sequences such as the alternating Fibonacci numbers, triangular numbers, and powers of two.
TL;DR: In this paper, it was shown that every connected trivalent G-symmetric graph of type Σ is a near polygonal graph with respect to two G-orbits on cycles of Σ.
Abstract: A connected graph Σ of girth at least four is called a near n-gonal graph with respect to E, where n ≥ 4 is an integer, if E is a set of n-cycles of Σ such that every path of length two is contained in a unique member of E. It is well known that connected trivalent symmetric graphs can be classified into seven types. In this note we prove that every connected trivalent G-symmetric graph \({\Sigma
eq K_4}\) of type \({G^1_2}\) is a near polygonal graph with respect to two G-orbits on cycles of Σ. Moreover, we give an algorithm for constructing the unique cycle in each of these G-orbits containing a given path of length two.
TL;DR: In this article, the determinant of a Hankel matrix whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes if and only if n is composite.
Abstract: We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes if and only if n is composite. If the dimension is a prime p, then the determinant evaluates to a polynomial of degree p − 1 which is the product of a power of p and the generating polynomial of the partial sums of Legendre symbols. The sign of the determinant is determined by the quadratic character of −1 modulo p. The proof of the evaluation makes use of elementary properties of Legendre symbols, quadratic Gauss sums, and orthogonality of trigonometric functions.
TL;DR: In this article, a bijection between permutations in Sn that avoid 3412 and contain exactly one 321 pattern was provided, where the permutation in Sn+1 that avoids 321 and contains exactly one 3412 pattern.
Abstract: We provide a bijection between the permutations in Sn that avoid 3412 and contain exactly one 321 pattern with the permutations in Sn+1 that avoid 321 and contain exactly one 3412 pattern. The enumeration of these classes is obtained from their classification via reduced decompositions. The results are extended to involutions in the above pattern classes using reduced decompositions reproducing a result of Egge.
TL;DR: In this paper, the problem of counting the number of ways to place nonattacking identical chess or fairy chess pieces in a rectangular strip of fixed height and variable width, as a function of the width, is converted to one of counting lattice points outside an affinographic hyperplane arrangement.
Abstract: The function that counts the number of ways to place nonattacking identical chess or fairy chess pieces in a rectangular strip of fixed height and variable width, as a function of the width, is a piecewise polynomial which is eventually a polynomial and whose behavior can be described in some detail. We deduce this by converting the problem to one of counting lattice points outside an affinographic hyperplane arrangement, which Forge and Zaslavsky solved by means of weighted integral gain graphs. We extend their work by developing both generating functions and a detailed analysis of deletion and contraction for weighted integral gain graphs. For chess pieces we find the asymptotic probability that a random configuration is nonattacking, and we obtain exact counts of nonattacking configurations of small numbers of queens, bishops, knights, and nightriders.