Showing papers in "Electronic Journal of Combinatorics in 1996"

Symmetry Breaking in Graphs

Abstract: A labeling of the vertices of a graph G, ` : V (G) !f1;:::;rg, is said to be r-distinguishing provided no automorphism of the graph preserves all of the vertex labels. The distinguishing number of a graph G, denoted by D(G), is the minimum r such that G has an r-distinguishing labeling. The distinguishing number of the complete graph on t vertices is t. In contrast, we prove (i) given any group i, there is a graph G such that Aut(G) » ia ndD(G )=2 ; (ii)D(G )= O(log(jAut(G)j)); (iii) if Aut(G) is abelian, then D(G) • 2; (iv) if Aut(G) is dihedral, then D(G) • 3; and (v) If Aut(G) » S4 ,t hen either D(G )= 2 orD(G) = 4. Mathematics Subject Classiflcation 05C,20B,20F,68R

Evaluations of $k$-fold Euler/Zagier sums: a compendium of results for arbitrary $k$

Abstract: Euler sums (also called Zagier sums) occur within the context of knot theory and quantum field theory. There are various conjectures related to these sums whose incompletion is a sign that both the mathematics and physics communities do not yet completely understand the field. Here, we assemble results for Euler/Zagier sums (also known as multidimensional zeta/harmonic sums) of arbitrary depth, including sign alternations. Many of our results were obtained empirically and are apparently new. By carefully compiling and examining a huge data base of high precision numerical evaluations, we can claim with some confidence that certain classes of results are exhaustive. While many proofs are lacking, we have sketched derivations of all results that have so far been proved.

Parking Functions and Noncrossing Partitions

Abstract: A parking function is a sequence $(a_1,\dots,a_n)$ of positive integers such that, if $b_1\leq b_2\leq \cdots\leq b_n$ is the increasing rearrangement of the sequence $(a_1,\dots, a_n),$ then $b_i\leq i$. A noncrossing partition of the set $[n]=\{1,2,\dots,n\}$ is a partition $\pi$ of the set $[n]$ with the property that if $a

Evaluation of triple euler sums

Abstract: Let a;b;c be positive integers and deflne the so-called triple, double and single Euler sums by

Treillis et bases des groupes de Coxeter

Abstract: Finite lattices possess a basis, as well as a cobasis, from which the elements of the lattice can be recovered by sup or inf. We extend this construction to finite ordered sets, and then apply it to Coxeter groups, considered as ordered sets (Bruhat order). This amounts to embedding Coxeter groups into their enveloping lattices. These lattices are distributive in the cases of types An and Bn.

The Tutte polynomial of a graph, depth-first search, and simplicial complex partitions.

Abstract: One of the most important numerical quantities that can be computed from a graph G is the two-variable Tutte polynomial. Specializations of the Tutte polynomial count various objects associated with G, e.g., subgraphs, spanning trees, acyclic orientations, inversions and parking functions. We show that by partitioning certain simplicial complexes related to G into intervals, one can provide combinatorial demonstrations of these results. One of the primary tools for providing such a partition is depth-first search.

Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes

Abstract: Bjorner and Wachs generalized the deflnition of shellability by dropping the as- sumption of purity; they also introduced the h-triangle, a doubly-indexed generaliza- tion of the h-vector which is combinatorially signiflcant for nonpure shellable com- plexes. Stanley subsequently deflned a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satisfles algebraic conditions that generalize the Cohen-Macaulay conditions for pure complexes, so that a nonpure shellable complex is sequentially Cohen-Macaulay. We show that algebraic shifting preserves the h-triangle of a simplicial complex K if and only if K is sequentially Cohen-Macaulay. This generalizes a result of Kalai's for the pure case. Immediate consequences include that nonpure shellable complexes and sequentially Cohen-Macaulay complexes have the same set of possible h-triangles. 1991 Mathematics Subject Classiflcation: Primary 06A08; Secondary 52B05.

Repeated Patterns of Dense Packings of Equal Disks in a Square

Abstract: We examine sequences of dense packings of $n$ congruent non-overlapping disks inside a square which follow specific patterns as $n$ increases along certain values, $n = n(1), n(2),... n(k),...$. Extending and improving previous work of Nurmela and Ostergard where previous patterns for $n = n(k)$ of the form $ k^2$, $ k^2-1$, $k^2-3$, $k(k+1)$, and $4k^2+k$ were observed, we identify new patterns for $n = k^2-2$ and $n = k^2+ \lfloor k/2 \rfloor$. We also find denser packings than those in Nurmela and Ostergard for $n =$21, 28, 34, 40, 43, 44, 45, and 47. In addition, we produce what we conjecture to be optimal packings for $n =$51, 52, 54, 55, 56, 60, and 61. Finally, for each identified sequence $n(1), n(2),... n(k),...$ which corresponds to some specific repeated pattern, we identify a threshold index $k_0$, for which the packing appears to be optimal for $k \le k_0$, but for which the packing is not optimal (or does not exist) for $k > k_0$.

The three dimensional polyominoes of minimal area

Abstract: The set of the three dimensional polyominoes of minimal area and of volume $n$ contains a polyomino which is the union of a quasicube $j\times (j+\delta)\times (j+\theta)$, $\delta,\theta\in\{0,1\}$, a quasisquare $l\times (l+\epsilon)$, $\epsilon\in\{0,1\}$, and a bar $k$. This shape is naturally associated to the unique decomposition of $n=j(j+\delta)(j+\theta)+l(l+\epsilon)+k$ as the sum of a maximal quasicube, a maximal quasisquare and a bar. For $n$ a quasicube plus a quasisquare, or a quasicube minus one, the minimal polyominoes are reduced to these shapes. The minimal area is explicitly computed and yields a discrete isoperimetric inequality. These variational problems are the key for finding the path of escape from the metastable state for the three dimensional Ising model at very low temperatures. The results and proofs are illustrated by a lot of pictures.

Dodgson's Determinant-Evaluation Rule Proved by Two-Timing Men and Women

Abstract: I give a bijective proof of the Reverend Charles Lutwidge Dodgson's Rule: $$ \det \left [ (a_{i,j})_{ {1 \leq i \leq n } \atop {1 \leq j \leq n }} \right ] \cdot \det \left [ (a_{i,j})_{ {2 \leq i \leq n-1 } \atop {2 \leq j \leq n-1 }} \right ] \, = $$ $$ \det \left [ (a_{i,j})_{ {1 \leq i \leq n-1 } \atop {1 \leq j \leq n-1 }} \right ] \cdot \det \left [ (a_{i,j})_{ {2 \leq i \leq n } \atop {2 \leq j \leq n }} \right ] \,-\, \det \left [ (a_{i,j})_{ {1 \leq i \leq n-1 } \atop {2 \leq j \leq n }} \right ] \cdot \det \left [ (a_{i,j})_{ {2 \leq i \leq n } \atop {1 \leq j \leq n-1 }} \right ]\quad . $$

Balanced Gray Codes

Abstract: It is shown that balanced $n$-bit Gray codes can be constructed for all positive integers $n$. A balanced Gray code is one in which the bit changes are distributed as equally as possible among the bit positions. The strategy used is to prove the existence of a certain subsequence which will allow successful use of the construction proposed by Robinson and Cohn in 1981. Although Wagner and West proved in 1991 that balanced Gray code schemes exist when $n$ is a power of 2, the question for general $n$ has remained open since 1980 when it first attracted attention.

Improved coverings of a square with six and eight equal circles

Abstract: In a recent article, Tarnai and Gaspar used computer simulations to find thin coverings of a square with up to ten equal circles. We will give improved coverings with six and eight circles and a new, thin covering with eleven circles, found by the use of simulated annealing. Furthermore, we present a combinatorial method for constructing lower bounds for the optimal covering radius.

Generating Functions and Generalized Dedekind Sums

Abstract: We study sums of the form $\sum_\zeta R(\zeta)$, where $R$ is a rational function and the sum is over all $n$th roots of unity $\zeta$ (often with $\zeta =1$ excluded). We call these generalized Dedekind sums, since the most well-known sums of this form are Dedekind sums. We discuss three methods for evaluating such sums: The method of factorization applies if we have an explicit formula for $\prod_\zeta (1-xR(\zeta))$. Multisection can be used to evaluate some simple, but important sums. Finally, the method of partial fractions reduces the evaluation of arbitrary generalized Dedekind sums to those of a very simple form.

Hypergeometric series acceleration via the wz method

Abstract: Based on the WZ method, some series acceleration formulas are given. These formulas allow us to write down an inflnite family of parametrized identities from any given identity of WZ type. Further, this family, in the case of the Zeta function, gives rise to many accelerated expressions for ‡(3).

Packing Graphs: The Packing Problem Solved

Abstract: For every fixed graph $H$, we determine the $H$-packing number of $K_n$, for all $n > n_0(H)$ We prove that if $h$ is the number of edges of $H$, and $gcd(H)=d$ is the greatest common divisor of the degrees of $H$, then there exists $n_0=n_0(H)$, such that for all $n > n_0$, $$ P(H,K_n)=\lfloor {{dn}\over{2h}} \lfloor {{n-1}\over{d}} \rfloor \rfloor, $$ unless $n = 1 \bmod d$ and $n(n-1)/d = b \bmod (2h/d)$ where $1 \leq b \leq d$, in which case $$ P(H,K_n)=\lfloor {{dn}\over{2h}} \lfloor {{n-1}\over{d}} \rfloor \rfloor - 1 $$ Our main tool in proving this result is the deep decomposition result of Gustavsson

Combinatorial Game Theory Foundations Applied to Digraph Kernels

Abstract: Known complexity facts: the decision problem of the existence of a kernel in a digraph $G=(V,E)$ is NP-complete; if all of the cycles of $G$ have even length, then $G$ has a kernel; and the question of the number of kernels is $\#$P-complete even for this restricted class of digraphs. In the opposite direction, we construct game theory tools, of independent interest, concerning strategies in the presence of draw positions, to show how to partition $V$, in $O(|E|)$ time, into $3$ subsets $S_1,S_2,S_3$, such that $S_1$ lies in all the kernels; $S_2$ lies in the complements of all the kernels; and on $S_3$ the kernels may be nonunique. Thus, in particular, digraphs with a "large" number of kernels are those in which $S_3$ is "large"; possibly $S_1=S_2=\emptyset$. We also show that $G$ can be decomposed, in $O(|E|)$ time, into two induced subgraphs $G_1$, with vertex-set $S_1\cup S_2$, which has a unique kernel; and $G_2$, with vertex-set $S_3$, such that any kernel $K$ of $G$ is the union of the kernel of $G_1$ and a kernel of $G_2$. In particular, $G$ has no kernel if and only if $G_2$ has none. Our results hold even for some classes of infinite digraphs.

From Recursions to Asymptotics: On Szekeres' Formula for the Number of Partitions

Abstract: We give a new proof of Szekeres’ formula for P(n,k), the number of partitions of the integer n having k or fewer positive parts. Our proof is based on the recursion satisfied by P(n,k) and Taylor’s formula. We make no use of the Cauchy integral formula or any complex variables. The derivation is presented as a step-by-step procedure, to facilitate its application in other situations. As corollaries we obtain the main term of the Hardy-Ramanujan formulas for p(n) = the number of unrestricted partitions of n, and for q(n) = the number of partitions of n into distinct parts.

An improved tableau criterion for Bruhat order.

Abstract: To decide whether two permutations are comparable in Bruhat order of Sn with the well-known tableau criterion requires i n ¢ comparisons of entries in certain sorted arrays.

Some Natural Bigraded $S_n$-Modules

Abstract: We construct for each $\mu\vdash n $ a bigraded $S_n$-module $\mathbf{H}_\mu$ and conjecture that its Frobenius characteristic $C_{\mu}(x;q,t)$ yields the Macdonald coefficients $K_{\lambda\mu}(q,t)$. To be precise, we conjecture that the expansion of $C_{\mu}(x;q,t)$ in terms of the Schur basis yields coefficients $C_{\lambda\mu}(q,t)$ which are related to the $K_{\lambda\mu}(q,t)$ by the identity $C_{\lambda\mu}(q,t)=K_{\lambda\mu}(q,1/t)t^{n(\mu )}$. The validity of this would give a representation theoretical setting for the Macdonald basis $\{ P_\mu(x;q,t)\}_\mu$ and establish the Macdonald conjecture that the $K_{\lambda\mu}(q,t)$ are polynomials with positive integer coefficients. The space $\mathbf{H}_\mu$ is defined as the linear span of derivatives of a certain bihomogeneous polynomial $\Delta_\mu(x,y)$ in the variables $x_1,x_2,\ldots ,x_n$, $y_1,y_2,\ldots ,y_n$. On the validity of our conjecture $\mathbf{H}_\mu$ would necessarily have $n!$ dimension. We refer to the latter assertion as the $n!$-conjecture. Several equivalent forms of this conjecture will be discussed here together with some of their consequences. In particular, we derive that the polynomials $C_{\lambda\mu}(q,t)$ have a number of basic properties in common with the coefficients $\tilde{K}_{\lambda\mu}(q,t)=K_{\lambda\mu}(q,1/t)t^{n(\mu )}$. For instance, we show that $C_{\lambda\mu}(0,t)=\tilde{K}_{\lambda\mu}(0,t)$, $C_{\lambda\mu}(q,0)=\tilde{K}_{\lambda\mu}(q,0)$ and show that on the $n!$ conjecture we must also have the equalities $C_{\lambda\mu}(1,t)=\tilde{K}_{\lambda\mu}(1,t)$ and $C_{\lambda\mu}(q,1)=\tilde{K}_{\lambda\mu}(q,1)$. The conjectured equality $C_{\lambda\mu}(q,t)=K_{\lambda\mu}(q,1/t)t^{n(\mu )}$ will be shown here to hold true when $\lambda$ or $\mu$ is a hook. It has also been shown (see [9]) when $\mu$ is a $2$-row or $2$-column partition and in [18] when $\mu$ is an augmented hook.

A New Construction for Cancellative Families of Sets

Abstract: Following [2], we say a family, $H$, of subsets of a $n$-element set is cancellative if $A \cup B = A \cup C$ implies $B =C$ when $A, B, C \in H$. We show how to construct cancellative families of sets with $c 2^{.54797n}$ elements. This improves the previous best bound $c 2^{.52832n}$ and falsifies conjectures of Erdos and Katona [3] and Bollobas [1].

The Number of Knight's Tours Equals 33,439,123,484,294 — Counting with Binary Decision Diagrams

Abstract: The number of knight's tours, i.e. Hamiltonian circuits, on an $8 \times 8$ chessboard is computed with decision diagrams which turn out to be a useful tool for counting problems.

On cycles in the coprime graph of integers

Abstract: In this paper we study cycles in the coprime graph of integers. We denote byf(n;k) the number of positive integers mnwith a prime factor among the flrst k primes. We show that there exists a constant c such that if A‰f 1 ; 2;:::;ngwithjAj >f (n; 2) (if 6jn then f(n; 2) = 2 n), then the coprime graph induced by A not only contains a triangle, but also a cycle of length 2l + 1 for every positive integer lcn.

A Purely Combinatorial Proof of the Hadwiger Debrunner (p;q) Conjecture

Abstract: Af amily of sets has the (p;q) property if among any p members of the family some q have a nonempty intersection. The authors have proved that for every p‚ q‚ d +1 there is a c = c(p;q;d) <1 such that for every familyF of compact, convex sets in R d which has the (p;q) property there is a set of at most c points in R d that intersects each member ofF, thus settling an old problem of Hadwiger and Debrunner. Here we present a purely combinatorial proof of this result. AMS Subject Classiflcation: 52A35

A Generalization of Gosper's Algorithm to Bibasic Hypergeometric Summation

Abstract: An algebraically motivated generalization of Gosper's algorithm to indefinite bibasic hypergeometric summation is presented. In particular, it is shown how Paule's concept of greatest factorial factorization of polynomials can be extended to the bibasic case. It turns out that most of the bibasic hypergeometric summation identities from literature can be proved and even found this way. A Mathematica implementation of the algorithm is available from the author.

Descendants in heap ordered trees or a triumph of computer algebra

Abstract: A heap ordered tree with n nodes (\size n") is a planted plane tree together with a bijection from the nodes to the set f1;:::;ng which is monotonically increasing when going from the root to the leaves. We consider the number of descendants of the node j in a (random) heap ordered tree of size n j. Precise expressions are derived for the probability distribution and all (factorial) moments. AMS Subject Classication. 05A15 (primary) 05C05 (secondary)

Playing Nim on a Simplicial Complex

Abstract: We introduce a generalization of the classical game of Nim by placing the piles on the vertices of a simplicial complex and allowing a move to aect the piles on any set of vertices that forms a face of the complex. Under certain conditions on the complex we present a winning strategy. These conditions are satised, for instance, when the simplicial complex consists of the independent sets of a binary matroid. Moreover, we study four operations on a simplicial complex under which games on the complex behave nicely. We also consider particular complexes that correspond to natural generalizations of classical Nim.

A Simple Method for Constructing Small Cubic Graphs of Girths 14, 15, and 16

Abstract: A method for constructing cubic graphs with girths in the range 13 to 16 is described. The method is used to construct the smallest known cubic graphs for girths 14, 15 and 16.

Random walks on generating sets for finite groups

Abstract: We analyze a certain random walk on the cartesian product $G^n$ of a finite group $G$ which is often used for generating random elements from $G$. In particular, we show that the mixing time of the walk is at most $c_r n^2 \log n$ where the constant $c_r$ depends only on the order $r$ of $G$.

Escher's Combinatorial Patterns

Abstract: It is a little-known fact that M.C. Escher posed and answered some combinatorial questions about patterns produced in an algorithmic way. We report on his explorations, indicate how close he came to the correct solutions, and pose an analagous problem in 3 dimensions.