# Vol. 11 no. 1

### 1. The distribution of ascents of size d or more in compositions

A composition of a positive integer n is a finite sequence of positive integers a(1), a(2), ..., a(k) such that a(1) + a(2) + ... + a(k) = n. Let d be a fixed nonnegative integer. We say that we have an ascent of size d or more if a(i+1) >= a(i) + d. We determine the mean, variance and limiting distribution of the number of ascents of size d or more in the set of compositions of n. We also study the average size of the greatest ascent over all compositions of n.
Section: Combinatorics

### 2. On the length of shortest 2-collapsing words

Given a word w over a finite alphabet Sigma and a finite deterministic automaton A = < Q,Sigma,delta >, the inequality vertical bar delta(Q,w)vertical bar <= vertical bar Q vertical bar - k means that under the natural action of the word w the image of the state set Q is reduced by at least k states. The word w is k-collapsing (k-synchronizing) if this inequality holds for any deterministic finite automaton ( with k + 1 states) that satisfies such an inequality for at least one word. We prove that for each alphabet Sigma there is a 2-collapsing word whose length is vertical bar Sigma vertical bar(3)+6 vertical bar Sigma vertical bar(2)+5 vertical bar Sigma vertical bar/2. Then we produce shorter 2-collapsing and 2-synchronizing words over alphabets of 4 and 5 letters.
Section: Automata, Logic and Semantics

### 3. Centerpoint theorems for wedges

The Centerpoint Theorem states that, for any set S of n points in R(d), there exists a point p in R(d) such that every closed halfspace containing p contains at least [n/(d + 1)] points of S. We consider generalizations of the Centerpoint Theorem in which halfspaces are replaced with wedges (cones) of angle alpha. In R(2), we give bounds that are tight for all values of ff and give an O(n) time algorithm to find a point satisfying these bounds. We also give partial results for R(3) and, more generally, R(d).

### 4. String pattern avoidance in generalized non-crossing trees

The problem of string pattern avoidance in generalized non-crossing trees is studied. The generating functions for generalized non-crossing trees avoiding string patterns of length one and two are obtained. The Lagrange inversion formula is used to obtain the explicit formulas for some special cases. A bijection is also established between generalized non-crossing trees with special string pattern avoidance and little Schr ̈oder paths.
Section: Combinatorics

### 5. Long cycles in hypercubes with distant faulty vertices

In this paper, we study long cycles in induced subgraphs of hypercubes obtained by removing a given set of faulty vertices such that every two faults are distant. First, we show that every induced subgraph of Q(n) with minimum degree n - 1 contains a cycle of length at least 2(n) - 2(f) where f is the number of removed vertices. This length is the best possible when all removed vertices are from the same bipartite class of Q(n). Next, we prove that every induced subgraph of Q(n) obtained by removing vertices of some given set M of edges of Q(n) contains a Hamiltonian cycle if every two edges of M are at distance at least 3. The last result shows that the shell of every linear code with odd minimum distance at least 3 contains a Hamiltonian cycle. In all these results we obtain significantly more tolerable faulty vertices than in the previously known results. We also conjecture that every induced subgraph of Q(n) obtained by removing a balanced set of vertices with minimum distance at least 3 contains a Hamiltonian cycle.
Section: Graph and Algorithms

### 6. Enumeration of words by the sum of differences between adjacent letters

We consider the sum u of differences between adjacent letters of a word of n letters, chosen uniformly at random from a given alphabet. This paper obtains the enumerating generating function for the number of such words with respect to the sum u, as well as explicit formulas for the mean and variance of u.
Section: Combinatorics

### 7. Self-complementing permutations of k-uniform hypergraphs

A k-uniform hypergraph H = ( V; E) is said to be self-complementary whenever it is isomorphic with its complement (H) over bar = ( V; ((V)(k)) - E). Every permutation sigma of the set V such that sigma(e) is an edge of (H) over bar if and only if e is an element of E is called self-complementing. 2-self-comlementary hypergraphs are exactly self complementary graphs introduced independently by Ringel ( 1963) and Sachs ( 1962). <br> For any positive integer n we denote by lambda(n) the unique integer such that n = 2(lambda(n)) c, where c is odd. <br> In the paper we prove that a permutation sigma of [1, n] with orbits O-1,..., O-m O m is a self-complementing permutation of a k-uniform hypergraph of order n if and only if there is an integer l >= 0 such that k = a2(l) + s, a is odd, 0 <= s <= 2(l) and the following two conditions hold: <br> (i)n = b2(l+1) + r,r is an element of {0,..., 2(l) - 1 + s}, and <br> (ii) Sigma(i:lambda(vertical bar Oi vertical bar)<= l) vertical bar O-i vertical bar <= r. <br> For k = 2 this result is the very well known characterization of self-complementing permutation of graphs given by Ringel and Sachs.
Section: Graph and Algorithms

### 8. The Eulerian distribution on centrosymmetric involutions

We present an extensive study of the Eulerian distribution on the set of centrosymmetric involutions, namely, involutions in S(n) satisfying the property sigma(i) + sigma(n + 1 - i) = n + 1 for every i = 1 ... n. We find some combinatorial properties for the generating polynomial of such distribution, together with an explicit formula for its coefficients. Afterwards, we carry out an analogous study for the subset of centrosymmetric involutions without fixed points.
Section: Combinatorics

### 9. Number of connected spanning subgraphs on the Sierpinski gasket

We study the number of connected spanning subgraphs f(d,b) (n) on the generalized Sierpinski gasket SG(d,b) (n) at stage n with dimension d equal to two, three and four for b = 2, and layer b equal to three and four for d = 2. The upper and lower bounds for the asymptotic growth constant, defined as zSG(d,b) = lim(v ->infinity) ln f(d,b)(n)/v where v is the number of vertices, on SG(2,b) (n) with b = 2, 3, 4 are derived in terms of the results at a certain stage. The numerical values of zSG(d,b) are obtained.
Section: Combinatorics

### 10. Asymptotic enumeration on self-similar graphs with two boundary vertices

We study two graph parameters, namely the number of spanning forests and the number of connected subgraphs, for self-similar graphs with exactly two boundary vertices. In both cases, we determine the general behavior for these and related auxiliary quantities by means of polynomial recurrences and a careful asymptotic analysis. It turns out that the so-called resistance scaling factor of a graph plays an essential role in both instances, a phenomenon that was previously observed for the number of spanning trees. Several explicit examples show that our findings are likely to hold in an even more general setting.
Section: Combinatorics

### 11. On-line extensible bin packing with unequal bin sizes

In the extensible bin packing problem we are asked to pack a set of items into a given number of bins, each with an original size. However, the original bin sizes can be extended if necessary. The goal is to minimize the total size of the bins. We consider the problem with unequal (original) bin sizes and give the complete analysis on a list scheduling algorithm (LS). Namely we present tight bounds of LS for every collection of original bin sizes and every number of bins. We further show better on-line algorithms for the two-bin case and the three-bin case. Interestingly, it is proved that the on-line algorithms have better competitive ratios for unequal bins than for equal bins. Some variants of the problem are also discussed.
Section: Analysis of Algorithms

### 12. Independent sets in (P₆, diamond)-free graphs

We prove that on the class of (P6,diamond)-free graphs the Maximum-Weight Independent Set problem and the Minimum-Weight Independent Dominating Set problem can be solved in polynomial time.
Section: Graph and Algorithms

### 13. Diophantine Approximation, Ostrowski Numeration and the Double-Base Number System

A partition of $x > 0$ of the form $x = \sum_i 2^{a_i}3^{b_i}$ with distinct parts is called a double-base expansion of $x$. Such a representation can be obtained using a greedy approach, assuming one can efficiently compute the largest \mbox{$\{2,3\}$-integer}, i.e., a number of the form $2^a3^b$, less than or equal to $x$. In order to solve this problem, we propose an algorithm based on continued fractions in the vein of the Ostrowski number system, we prove its correctness and we analyse its complexity. In a second part, we present some experimental results on the length of double-base expansions when only a few iterations of our algorithm are performed.
Section: Analysis of Algorithms